{"id":216,"date":"2025-04-09T17:29:20","date_gmt":"2025-04-09T17:29:20","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/7-5-matrices-and-matrix-operations-college-algebra-2e-openstax\/"},"modified":"2025-09-05T15:31:45","modified_gmt":"2025-09-05T15:31:45","slug":"7-5-matrices-and-matrix-operations","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/7-5-matrices-and-matrix-operations\/","title":{"raw":"7.5 Matrices and Matrix Operations","rendered":"7.5 Matrices and Matrix Operations"},"content":{"raw":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\r\n<div id=\"page_508a4d4e-0136-4c6c-87b8-e210022d69b4\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Find the sum and difference of two matrices.<\/li>\r\n \t<li>Find scalar multiples of a matrix.<\/li>\r\n \t<li>Find the product of two matrices.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n[caption id=\"attachment_1723\" align=\"aligncenter\" width=\"297\"]<img class=\"size-medium wp-image-1723\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.5-fig-1-297x300.webp\" alt=\"\" width=\"297\" height=\"300\" \/> Figure 1 (credit: \"SD Dirk,\" Flickr)[\/caption]\r\n<p id=\"fs-id1165137862059\">Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season. Table 1 shows the needs of both teams.<\/p>\r\n\r\n<div id=\"Table_09_05_01\" class=\"os-table\">\r\n<table class=\"grid\" style=\"height: 96px; width: 356px;\" data-id=\"Table_09_05_01\"><caption>Table 1<\/caption>\r\n<thead>\r\n<tr style=\"height: 24px;\">\r\n<th style=\"height: 24px; width: 107.733px;\" scope=\"col\"><\/th>\r\n<th style=\"height: 24px; width: 122.233px;\" scope=\"col\" data-align=\"center\">Wildcats<\/th>\r\n<th style=\"height: 24px; width: 104.033px;\" scope=\"col\" data-align=\"center\">Mud Cats<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 24px;\">\r\n<td style=\"height: 24px; width: 107.733px;\" data-align=\"center\"><strong>Goals<\/strong><\/td>\r\n<td style=\"height: 24px; width: 122.233px;\" data-align=\"center\">6<\/td>\r\n<td style=\"height: 24px; width: 104.033px;\" data-align=\"center\">10<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\">\r\n<td style=\"height: 24px; width: 107.733px;\" data-align=\"center\"><strong>Balls<\/strong><\/td>\r\n<td style=\"height: 24px; width: 122.233px;\" data-align=\"center\">30<\/td>\r\n<td style=\"height: 24px; width: 104.033px;\" data-align=\"center\">24<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\">\r\n<td style=\"height: 24px; width: 107.733px;\" data-align=\"center\"><strong>Jerseys<\/strong><\/td>\r\n<td style=\"height: 24px; width: 122.233px;\" data-align=\"center\">14<\/td>\r\n<td style=\"height: 24px; width: 104.033px;\" data-align=\"center\">20<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\">A goal costs $300; a ball costs $10; and a jersey costs $30. How can we find the total cost for the equipment needed for each team? In this section, we discover a method in which the data in the soccer equipment table can be displayed and used for calculating other information. Then, we will be able to calculate the cost of the equipment.<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165132962023\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Finding the Sum and Difference of Two Matrices<\/h2>\r\n<p id=\"fs-id1165134430367\">To solve a problem like the one described for the soccer teams, we can use a <span id=\"term-00005\" class=\"no-emphasis\" data-type=\"term\">matrix<\/span>, which is a rectangular array of numbers. A <span id=\"term-00006\" class=\"no-emphasis\" data-type=\"term\">row<\/span> in a matrix is a set of numbers that are aligned horizontally. A <span id=\"term-00007\" class=\"no-emphasis\" data-type=\"term\">column<\/span> in a matrix is a set of numbers that are aligned vertically. Each number is an <span id=\"term-00008\" class=\"no-emphasis\" data-type=\"term\">entry<\/span>, sometimes called an element, of the matrix. Matrices (plural) are enclosed in [ ] or ( ), and are usually named with capital letters. For example, three matrices named\u00a0<em>A, B<\/em>\u00a0and\u00a0<em>C<\/em> are shown below.<\/p>\r\n<p style=\"text-align: center;\">[latex] A = \\begin{bmatrix} 1 &amp; 2 \\\\ 3 &amp; 4 \\end{bmatrix}, B = \\begin{bmatrix} 1 &amp; 2 &amp; 7 \\\\ 0 &amp; -5 &amp; 6 \\\\ 7 &amp; 8 &amp; 2 \\end{bmatrix}, C = \\begin{bmatrix} -1 &amp; 3 \\\\ 0 &amp; 2 \\\\ 3 &amp; 1 \\end{bmatrix} [\/latex]<\/p>\r\n\r\n<section id=\"fs-id1165132938226\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Describing Matrices<\/h3>\r\n<p id=\"fs-id1165134060293\">A matrix is often referred to by its size or dimensions: [latex] m\\times n [\/latex] indicating <em>m<\/em> rows and <em>n<\/em> columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix <em>A<\/em> identified as [latex] a_{ij}, [\/latex] we look for the entry in row <em>i<\/em>, column <em>j<\/em>. In matrix <em>A<\/em>, shown below, the entry in row 2, column 3 is [latex] a_{23}. [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] A = \\begin{bmatrix} a_{11} &amp; a_{12} &amp; a_{13} \\\\ a_{21} &amp; a_{22} &amp; a_{23} \\\\ a_{31} &amp; a_{32} &amp; a_{33} \\end{bmatrix} [\/latex]<\/p>\r\n<p id=\"fs-id1165137629048\">A square matrix is a matrix with dimensions [latex] n\\times n [\/latex] meaning that it has the same number of rows as columns. The [latex] 3\\times 3 [\/latex] matrix above is an example of a square matrix.<\/p>\r\n<p id=\"fs-id1165137400185\">A row matrix is a matrix consisting of one row with dimensions [latex] 1\\times n. [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{bmatrix} a_{11} &amp; a_{12} &amp; a_{13} \\end{bmatrix} [\/latex]<\/p>\r\n<p id=\"fs-id1165137475278\">A column matrix is a matrix consisting of one column with dimensions [latex] m\\times 1. [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{bmatrix} a_{11} \\\\ a_{21} \\\\ a_{31} \\end{bmatrix} [\/latex]<\/p>\r\n<p id=\"fs-id1165135307784\">A matrix may be used to represent a system of equations. In these cases, the numbers represent the coefficients of the variables in the system. Matrices often make solving systems of equations easier because they are not encumbered with variables. We will investigate this idea further in the next section, but first we will look at basic matrix operations.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Matrices<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA <strong>matrix <\/strong>is a rectangular array of numbers that is usually named by a capital letter:\u00a0<em>A, B, C <\/em>and so on. Each entry in a matrix is referred to as [latex] a_{ij}, [\/latex] such that <em>i<\/em> represents the row and\u00a0<em>j<\/em> represents the column. Matrices are often referred to by their dimensions: [latex] m\\times n [\/latex] indicating <em>m\u00a0<\/em>rows and\u00a0<em>n\u00a0<\/em>columns.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/section><\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 1: Finding the Dimensions of the Given Matrix and Locating Entries<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven matrix\u00a0<em>A:<\/em>\r\n\r\n(a) What are the dimensions of matrix\u00a0<em>A?<\/em>\r\n\r\n(b) What are the entries at [latex] a_{31} [\/latex] and [latex] a_{22} [\/latex]\r\n<p style=\"text-align: center;\">[latex] A=\\begin{bmatrix} 2 &amp; 1 &amp; 0 \\\\ 2 &amp; 4 &amp; 7 \\\\ 3 &amp; 1 &amp; -2 \\end{bmatrix} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>(a) The dimensions are [latex] 3\\times 3 [\/latex] because there are three rows and three columns.\r\n\r\n(b) Entry [latex] a_{31} [\/latex] is the number at row 3, column 1, which is 3. The entry [latex] a_{22} [\/latex] is the number at row 2, column 2, which is 4. Remember, the row comes first, then the column.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<section id=\"fs-id1165135499659\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Adding and Subtracting Matrices<\/h3>\r\n<p id=\"fs-id1165137602057\">We use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries.<\/p>\r\n<p id=\"fs-id1165137590185\">In order to do this, the entries must correspond. Therefore, <em data-effect=\"italics\">addition and subtraction of matrices is only possible when the matrices have the same dimensions<\/em>. We can add or subtract a [latex] 3\\times 3 [\/latex] matrix and another [latex] 3\\times 3 [\/latex] matrix, but we cannot add or subtract a [latex] 2\\times 3 [\/latex] matrix and a [latex] 3\\times 3 [\/latex] matrix because some entries in one matrix will not have a corresponding entry in the other matrix.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Adding and Subtracting Matrices<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven matrices\u00a0<em>A<\/em> and\u00a0<em>B<\/em> of like dimensions, addition and subtraction of\u00a0<em>A<\/em> and\u00a0<em>B\u00a0<\/em>will produce matrix\u00a0<em>C\u00a0<\/em>or matrix\u00a0<em>D <\/em>of the same dimension.\r\n<p style=\"text-align: center;\">[latex] A+B=C [\/latex] such that [latex] a_{ij}+b_{ij}=c_{ij} [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] A-B=D [\/latex] such that [latex] a_{ij}-b_{ij}=d_{ij} [\/latex]<\/p>\r\nMatrix addition is commutative.\r\n<p style=\"text-align: center;\">[latex] A+B=B+A [\/latex]<\/p>\r\nIt is also associative.\r\n<p style=\"text-align: center;\">[latex] (A+B)+C=A+(B+C) [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 2: Finding the Sum of Matrices<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the sum of <em>A<\/em> and <em>B<\/em>, given\r\n<p style=\"text-align: center;\">[latex] A = \\begin{bmatrix} a &amp; b \\\\ c &amp; d \\end{bmatrix} \\quad \\text{and} \\quad B = \\begin{bmatrix} e &amp; f \\\\ g &amp; h \\end{bmatrix} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Add corresponding entries.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} A+B &amp;=&amp; \\begin{bmatrix} a &amp; b \\\\ c &amp; d \\end{bmatrix}+\\begin{bmatrix} e &amp; f \\\\ g &amp; h \\end{bmatrix} \\\\ \\\\ &amp;=&amp; \\begin{bmatrix} a+e &amp; b+f \\\\ c+g &amp; d+h \\end{bmatrix} \\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3: Adding Matrix <em>A\u00a0<\/em>and Matrix\u00a0<em>B<\/em><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the sum of <em>A<\/em> and <em>B<\/em>.\r\n\r\n[latex] A = \\begin{bmatrix} 4 &amp; 1 \\\\ 3 &amp; 2 \\end{bmatrix} \\quad \\text{and} \\quad B = \\begin{bmatrix} 5 &amp; 9 \\\\ 0 &amp; 7 \\end{bmatrix} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Add corresponding entries. Add the entry in row 1, column 1, [latex] a_{11}, [\/latex] of matrix <em>A<\/em> to the entry in row 1, column 1, [latex] b_{11}, [\/latex] of <em>B<\/em>. Continue the pattern until all entries have been added.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} A+B &amp;=&amp; \\begin{bmatrix} 4 &amp; 1 \\\\ 3 &amp; 2 \\end{bmatrix}+\\begin{bmatrix} 5 &amp; 9 \\\\ 0 &amp; 7 \\end{bmatrix} \\\\ \\\\ &amp;=&amp; \\begin{bmatrix} 4+5 &amp; 1+9 \\\\ 3+0 &amp; 2+7 \\end{bmatrix} \\\\ \\\\ &amp;=&amp; \\begin{bmatrix} 9 &amp; 10 \\\\ 3 &amp; 9 \\end{bmatrix} \\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 4: Finding the Difference of Two Matrices<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the difference of <em>A<\/em> and <em>B<\/em>.\r\n\r\n[latex] A = \\begin{bmatrix} -2 &amp; 3 \\\\ 0 &amp; 1 \\end{bmatrix} \\quad \\text{and} \\quad B = \\begin{bmatrix} 8 &amp; 1 \\\\ 5 &amp; 4 \\end{bmatrix} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>We subtract the corresponding entries of each matrix.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} A+B &amp;=&amp; \\begin{bmatrix} -2 &amp; 3 \\\\ 0 &amp; 1 \\end{bmatrix}-\\begin{bmatrix} 8 &amp; 1 \\\\ 5 &amp; 4 \\end{bmatrix} \\\\ \\\\ &amp;=&amp; \\begin{bmatrix} -2-8 &amp; 3-1 \\\\ 0-5 &amp; 1-4 \\end{bmatrix} \\\\ \\\\ &amp;=&amp; \\begin{bmatrix} -10 &amp; 2 \\\\ -5 &amp; -3 \\end{bmatrix} \\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 5: Finding the Sum and Difference of Two 3x3 Matrices<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven <em>A<\/em> and <em>B<\/em>:\r\n\r\n(a) Find the sum.\r\n\r\n(b) Find the difference.\r\n\r\n[latex] A = \\begin{bmatrix} 2 &amp; -10 &amp; -2 \\\\ 14 &amp; 12 &amp; 10 \\end{bmatrix} \\quad \\text{and} \\quad B = \\begin{bmatrix} 6 &amp; 10 &amp; -2 \\\\ 0 &amp; =12 &amp; -4 \\\\ -5 &amp; 2 &amp; -2\\end{bmatrix} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>(a) Add the corresponding entries.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} A+B &amp;=&amp; \\begin{bmatrix} 2 &amp; -10 &amp; -2 \\\\ 14 &amp; 12 &amp; 10 \\\\ 4 &amp; -2 &amp; 2 \\end{bmatrix}+\\begin{bmatrix} 6 &amp; 10 &amp; -2 \\\\ 0 &amp; -12 &amp; -4 \\\\ -5 &amp; 2 &amp; -2 \\end{bmatrix} \\\\ \\\\ &amp;=&amp; \\begin{bmatrix} 2+6 &amp; -10+1- &amp; -2-2 \\\\ 14+0 &amp; 12-12 &amp; 10-4 \\\\ 4-5 &amp; -2+2 &amp; 2-2 \\end{bmatrix} \\\\ \\\\ &amp;=&amp; \\begin{bmatrix} 8 &amp; 0 &amp; -4 \\\\ 14 &amp; 0 &amp; 6 \\\\ -1 &amp; 0 &amp; 0 \\end{bmatrix} \\end{array} [\/latex]<\/p>\r\n(b) Subtract the corresponding entries.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} A+B &amp;=&amp; \\begin{bmatrix} 2 &amp; -10 &amp; -2 \\\\ 14 &amp; 12 &amp; 10 \\\\ 4 &amp; -2 &amp; 2 \\end{bmatrix}-\\begin{bmatrix} 6 &amp; 10 &amp; -2 \\\\ 0 &amp; -12 &amp; -4 \\\\ -5 &amp; 2 &amp; -2 \\end{bmatrix} \\\\ \\\\ &amp;=&amp; \\begin{bmatrix} 2-6 &amp; -10-10 &amp; -2+2 \\\\ 14-0 &amp; 12+12 &amp; 10+4 \\\\ 4+5 &amp; -2-2 &amp; 2+2 \\end{bmatrix} \\\\ \\\\ &amp;=&amp; \\begin{bmatrix} -4 &amp; -20 &amp; 0 \\\\ 14 &amp; 24 &amp; 14 \\\\ 9 &amp; -4 &amp; 4 \\end{bmatrix} \\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAdd matrix <em>A<\/em> and matrix <em>B<\/em>.\r\n<p style=\"text-align: center;\">[latex] A = \\begin{bmatrix} 2 &amp; 6 \\\\ 1 &amp; 0 \\\\ 1 &amp; -3 \\end{bmatrix} \\quad \\text{and} \\quad B = \\begin{bmatrix} 3 &amp; -2 \\\\ 1 &amp; 5 \\\\ -4 &amp; 3 \\end{bmatrix} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165134389759\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Finding Scalar Multiples of a Matrix<\/h2>\r\n<p id=\"fs-id1165137517260\">Besides adding and subtracting whole matrices, there are many situations in which we need to multiply a matrix by a constant called a scalar. Recall that a <span id=\"term-00014\" class=\"no-emphasis\" data-type=\"term\">scalar<\/span> is a real number quantity that has magnitude, but not direction. For example, time, temperature, and distance are scalar quantities. The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A <strong>scalar multiple<\/strong> is any entry of a matrix that results from scalar multiplication.<\/p>\r\n<p id=\"fs-id1165137686653\">Consider a real-world scenario in which a university needs to add to its inventory of computers, computer tables, and chairs in two of the campus labs due to increased enrollment. They estimate that 15% more equipment is needed in both labs. The school\u2019s current inventory is displayed in Table 2.<\/p>\r\n\r\n<div id=\"Table_09_05_02\" class=\"os-table\">\r\n<table class=\"grid\" style=\"height: 96px; width: 384px;\" data-id=\"Table_09_05_02\"><caption>Table 2<\/caption>\r\n<thead>\r\n<tr style=\"height: 24px;\">\r\n<th style=\"height: 24px; width: 169.733px;\" scope=\"col\"><\/th>\r\n<th style=\"height: 24px; width: 93.4167px;\" scope=\"col\" data-align=\"center\">Lab A<\/th>\r\n<th style=\"height: 24px; width: 98.85px;\" scope=\"col\" data-align=\"center\">Lab B<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 24px;\">\r\n<td style=\"height: 24px; width: 169.733px;\" data-align=\"center\"><strong>Computers<\/strong><\/td>\r\n<td style=\"height: 24px; width: 93.4167px;\" data-align=\"center\">15<\/td>\r\n<td style=\"height: 24px; width: 98.85px;\" data-align=\"center\">27<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\">\r\n<td style=\"height: 24px; width: 169.733px;\" data-align=\"center\"><strong>Computer Tables<\/strong><\/td>\r\n<td style=\"height: 24px; width: 93.4167px;\" data-align=\"center\">16<\/td>\r\n<td style=\"height: 24px; width: 98.85px;\" data-align=\"center\">34<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\">\r\n<td style=\"height: 24px; width: 169.733px;\" data-align=\"center\"><strong>Chairs<\/strong><\/td>\r\n<td style=\"height: 24px; width: 93.4167px;\" data-align=\"center\">16<\/td>\r\n<td style=\"height: 24px; width: 98.85px;\" data-align=\"center\">34<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\">\r\n\r\nConverting the data to a matrix, we have\r\n<p style=\"text-align: center;\">[latex] C_{2013}=\\begin{bmatrix} 15 &amp; 27 \\\\ 16 &amp; 34 \\\\ 16 &amp; 34 \\end{bmatrix} [\/latex]<\/p>\r\n<p id=\"fs-id1165134583398\">To calculate how much computer equipment will be needed, we multiply all entries in matrix <em>C<\/em> by 0.15.<\/p>\r\n<p style=\"text-align: center;\">[latex] (0.15)C_{2013}=\\begin{bmatrix} (0.15)15 &amp; (0.15)27 \\\\ (0.15)16 &amp; (0.15)34 \\\\ (0.15)16 &amp; (0.15)34 \\end{bmatrix}=\\begin{bmatrix} 2.25 &amp; 4.05 \\\\ 2.4 &amp; 5.1 \\\\ 2.4 &amp; 5.1 \\end{bmatrix} [\/latex]<\/p>\r\n<p id=\"fs-id1165134259236\">We must round up to the next integer, so the amount of new equipment needed is<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{bmatrix} 3 &amp; 5 \\\\ 3 &amp; 6 \\\\ 3 &amp; 6 \\end{bmatrix} [\/latex]<\/p>\r\n<p id=\"fs-id1165135169236\">Adding the two matrices as shown below, we see the new inventory amounts.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{bmatrix} 15 &amp; 27 \\\\ 16 &amp; 34 \\\\ 16 &amp; 34 \\end{bmatrix}+\\begin{bmatrix} 3 &amp; 5 \\\\ 3 &amp; 6 \\\\ 3 &amp; 6 \\end{bmatrix}=\\begin{bmatrix} 18 &amp; 32 \\\\ 19 &amp; 40 \\\\ 19 &amp; 40 \\end{bmatrix} [\/latex]<\/p>\r\n<p id=\"fs-id1165135487154\">This means<\/p>\r\n<p style=\"text-align: center;\">[latex] C_{2014}=\\begin{bmatrix} 18 &amp; 32 \\\\ 19 &amp; 40 \\\\ 19 &amp; 40 \\end{bmatrix} [\/latex]<\/p>\r\n<p id=\"fs-id1165135186654\">Thus, Lab A will have 18 computers, 19 computer tables, and 19 chairs; Lab B will have 32 computers, 40 computer tables, and 40 chairs.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Scalar Multiplication<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nScalar multiplication involves finding the product of a constant by each entry in the matrix. Given\r\n<p style=\"text-align: center;\">[latex] A=\\begin{bmatrix} a_{11} &amp; a_{12} \\\\ a_{21} &amp; a_{22}\u00a0 \\end{bmatrix} [\/latex]<\/p>\r\nthe scalar multiple [latex] cA [\/latex] is\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rl} cA=c\\begin{bmatrix} a_{11} &amp; a_{12} \\\\ a_{21} &amp; a_{22} \\end{bmatrix} \\\\ &amp;=&amp; \\begin{bmatrix} ca_{11} &amp; ca_{12} \\\\ ca_{21} &amp; ca_{22} \\end{bmatrix} \\end{array} [\/latex]<\/p>\r\nScalar multiplication is distributive. For the matrices <em>A<\/em>, <em>B<\/em>, and <em>C<\/em> with scalars <em>a<\/em> and <em>b<\/em>,\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rl} a(A+B) &amp;=&amp; aA+aB \\\\ (a=b)A &amp;=&amp; aA+bA \\end{array} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 6: Multiplying the Matrix by a Scalar<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nMultiply matrix <em>A<\/em> by the scalar 3.\r\n<p style=\"text-align: center;\">[latex] A=\\begin{bmatrix} 8 &amp; 1 \\\\ 5 &amp; 4 \\end{bmatrix} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Multiply each entry in <em>A<\/em> by the scalar 3.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rl} 3A &amp;=&amp; 3\\begin{bmatrix} 8 &amp; 1 \\\\ 5 &amp; 4 \\end{bmatrix} \\\\ &amp;=&amp; \\begin{bmatrix} 3\\cdot 8 &amp; 3\\cdot 1 \\\\ 3\\cdot 5 &amp; 3\\cdot 4\u00a0 \\end{bmatrix} \\\\ &amp;=&amp; \\begin{bmatrix} 24 &amp; 3 \\\\ 15 &amp; 12\u00a0 \\end{bmatrix} \\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven matrix <em>B<\/em> find <em>-2B<\/em> where\r\n<p style=\"text-align: center;\">[latex] B=\\begin{bmatrix} 4 &amp; 1 \\\\ 3 &amp; 2\u00a0 \\end{bmatrix} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 7: Finding the Sum of Scalar Multiples<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the sum [latex] 3A+2B [\/latex]\r\n<p style=\"text-align: center;\">[latex] A= \\begin{bmatrix} 1 &amp; -2 &amp; 0 \\\\ 0 &amp; -1 &amp; 2 \\\\ 4 &amp; 3 &amp; -6\u00a0 \\end{bmatrix}\u00a0 \\quad \\text{and} \\quad B= \\begin{bmatrix}\u00a0 -1 &amp; 2 &amp; 1 \\\\ 0 &amp; -3 &amp; 2 \\\\ 0 &amp; 1 &amp; -4 \\end{bmatrix} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>First, find <em>3A<\/em>, then <em>2B<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rl} 3A &amp;=&amp; \\begin{bmatrix} 3\\cdot 1 &amp; 3(-2) &amp; 3\\cdot 0 \\\\ 3\\cdot 0 &amp; 3(-1) &amp; 3\\cdot 2 \\\\ 3\\cdot 4 &amp; 3\\cdot 3 &amp; 3(-6)\u00a0 \\end{bmatrix} \\\\ &amp;=&amp; \\begin{bmatrix} 3 &amp; -6 &amp; 0 \\\\ 0 &amp; -3 &amp; 6 \\\\ 12 &amp; 9 &amp; 18\u00a0 \\end{bmatrix} \\\\ \\\\ 2B &amp;=&amp;\u00a0 \\begin{bmatrix} 2(-1) &amp; 2\\cdot 2 &amp; 2\\cdot 1 \\\\ 2\\cdot 0 &amp; 2(-3) &amp; 2\\cdot 2 \\\\ 2\\cdot 0 &amp; 2\\cdot 1 &amp; 2(-4)\u00a0 \\end{bmatrix} \\\\\u00a0 &amp;=&amp; \\begin{bmatrix} -2 &amp; 4 &amp; 2 \\\\ 0 &amp; -6 &amp; 4 \\\\ 0 &amp; 2 &amp; -8\u00a0 \\end{bmatrix} \\end{array} [\/latex]<\/p>\r\nNow, add [latex] 3A+2B [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rl} 3A+2B &amp;=&amp;\u00a0 \\begin{bmatrix} 3 &amp; -6 &amp; 0 \\\\ 0 &amp; -3 &amp; 6 \\\\ 12 &amp; 9 &amp; -18\u00a0 \\end{bmatrix}+ \\begin{bmatrix} -2 &amp; 4 &amp; 2 \\\\ 0 &amp; -6 &amp; 4 \\\\ 0 &amp; 2 &amp; -8\u00a0 \\end{bmatrix} \\\\ &amp;=&amp; \\begin{bmatrix} 3-2 &amp; -6+4 &amp; 0+2 \\\\ 0+0 &amp; -3-6 &amp; 6+4 \\\\ 12+0 &amp; 9+2 &amp; -18-8 \\end{bmatrix} \\\\ &amp;=&amp;\u00a0 \\begin{bmatrix} 1 &amp; -2 &amp; 2 \\\\ 0 &amp; -9 &amp; 10 \\\\ 12 &amp; 11 &amp; -26\u00a0 \\end{bmatrix}\u00a0 \\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165133073877\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Finding the Product of Two Matrices<\/h2>\r\n<p id=\"fs-id1165134371143\">In addition to multiplying a matrix by a scalar, we can multiply two matrices. Finding the product of two matricesis only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. If <em>A<\/em> is an [latex] m\\times r [\/latex] matrix and <em>B<\/em> is an [latex] r\\times n [\/latex] matrix, then the product matrix <em>AB<\/em> is an [latex] m\\times n [\/latex] matrix. For example, the product <em>AB<\/em> is possible because the number of columns in <em>A<\/em> is the same as the number of rows in <em>B<\/em> If the inner dimensions do not match, the product is not defined.<\/p>\r\n<span id=\"eip-id1165134546098\" data-type=\"media\" data-alt=\"\" data-display=\"block\"><img class=\"size-full wp-image-1728 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.5-matrix.webp\" alt=\"\" width=\"154\" height=\"93\" \/><\/span>\r\n<p id=\"fs-id1165135649448\">We multiply entries of <em>A<\/em> with entries of <em>B<\/em> according to a specific pattern as outlined below. The process of matrix multiplication becomes clearer when working a problem with real numbers.<\/p>\r\n<p id=\"fs-id1165131956653\">To obtain the entries in row <em>i<\/em> of <em>AB<\/em>, we multiply the entries in row <em>i<\/em> of <em>A<\/em> by column <em>j<\/em> in <em>B<\/em> and add. For example, given matrices <em>A<\/em> and <em>B<\/em> where the dimensions of <em>A<\/em> are [latex] 2\\times 3 [\/latex] and the dimensions of <em>B<\/em> are [latex] 3\\times 3 [\/latex] the product of <em>AB<\/em> will be a [latex] 2\\times 3 [\/latex] matrix.<\/p>\r\n<p style=\"text-align: center;\">[latex] A=\\begin{bmatrix} a_{11} &amp; a_{12} &amp; a_{13} \\\\ a _{21} &amp; a_{22} &amp; a_{23} \\end{bmatrix} \\quad \\text{and} \\quad B=\\begin{bmatrix} b_{11} &amp; b_{12} &amp; b_{13} \\\\ b_{21} &amp; b_{22} &amp; b_{23} \\\\ b_{31} &amp; b_{32} &amp; b_{33} \\end{bmatrix} [\/latex]<\/p>\r\n<p id=\"fs-id1165134279285\">Multiply and add as follows to obtain the first entry of the product matrix <em>AB<\/em><\/p>\r\n\r\n<ol id=\"fs-id1165134547418\" type=\"1\">\r\n \t<li>To obtain the entry in row 1, column 1 of <em>AB<\/em>, multiply the first row in <em>A<\/em> by the first column in <em>B<\/em> and add.\r\n[latex] \\begin{bmatrix} a_{11} &amp; a_{12} &amp; a_{13} \\end{bmatrix} \\begin{bmatrix} b_{11} \\\\ b_{21} \\\\ b_{31} \\end{bmatrix}=a_{11}\\cdot b_{11} + a_{12}\\cdot b_{21} + a_{13}\\cdot b_{31} [\/latex]<\/li>\r\n \t<li>To obtain the entry in row 1, column 2 of <em>AB<\/em>, multiply the first row of <em>A<\/em> by the second column in <em>B<\/em> and add.\r\n[latex] \\begin{bmatrix} a_{11} &amp; a_{12} &amp; a_{13} \\end{bmatrix}\\begin{bmatrix} b_{12} \\\\ b_{22} \\\\ b_{32} \\end{bmatrix}=a_{11}\\cdot b_{12} + a_{12}\\cdot b_{22} + a_{13}\\cdot b_{32} [\/latex]<\/li>\r\n \t<li>To obtain the entry in row 1, column 3 of <em>AB<\/em>, multiply the first row of <em>A<\/em> by the third column in <em>B<\/em> and add.\r\n[latex] \\begin{bmatrix} a_{11} &amp; a_{12} &amp; a_{13} \\end{bmatrix}\\begin{bmatrix} b_{13} \\\\ b_{23} \\\\ b_{33} \\end{bmatrix}=a_{11}\\cdot b_{13} + a_{12}\\cdot b_{23} + a_{13}\\cdot b_{33} [\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165134589491\">We proceed the same way to obtain the second row of <em>AB<\/em>. In other words, row 2 of <em>A<\/em> times column 1 of <em>B<\/em>; row 2 of <em>A<\/em> times column 2 of <em>B<\/em>; row 2 of <em>A<\/em> times column 3 <em>of<\/em> B. When complete, the product matrix will be<\/p>\r\n[latex] AB=\\begin{bmatrix} a_{11}\\cdot b_{11} + a_{12}\\cdot b_{21} + a_{13}\\cdot b_{31} &amp; a_{11}\\cdot b_{12} + a_{12}\\cdot b_{22} + a_{13}\\cdot b_{32} &amp; a_{11}\\cdot b_{13} + a_{12}\\cdot b_{23} + a_{13}\\cdot b_{33} \\\\ a_{21}\\cdot b_{11} + a_{22}\\cdot b_{21} + a_{23}\\cdot b_{31} &amp; a_{21}\\cdot b_{12} + a_{22}\\cdot b_{22} + a_{23}\\cdot b_{32} &amp; a_{21}\\cdot b_{13} + a_{22}\\cdot b_{23} + a_{23}\\cdot b_{33} \\end{bmatrix} [\/latex]\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Properties of Matrix Multiplication<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFor the matrices <em>A<\/em>, <em>B<\/em>, and <em>C<\/em> the following properties hold.\r\n<ul>\r\n \t<li>Matrix multiplication is associative: [latex] (AB)C=A(BC). [\/latex]<\/li>\r\n \t<li>Matrix multiplication is distributive: [latex] \\begin{array}{rl} C(A+B) &amp;=&amp; CA+CB, \\\\ (A+B)C &amp;=&amp; AC+BC. \\end{array} [\/latex]<\/li>\r\n<\/ul>\r\nNote that matrix multiplication is not commutative.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 8: Multiplying Two Matrices<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nMultiply matrix <em>A<\/em> and matrix <em>B<\/em>.\r\n\r\n[latex] A=\\begin{bmatrix} 1 &amp; 2 \\\\ 3 &amp; 4 \\end{bmatrix} \\quad \\text{and} \\quad B=\\begin{bmatrix} 5 &amp; 6 \\\\ 7 &amp; 8 \\end{bmatrix} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>First, we check the dimensions of the matrices. Matrix <em>A<\/em> has dimensions [latex] 2\\times 2 [\/latex] and matrix <em>B<\/em> has dimensions [latex] 2\\times 2 [\/latex] The inner dimensions are the same so we can perform the multiplication. The product will have the dimensions [latex] 2\\times 2. [\/latex]\r\n\r\nWe perform the operations outlined previously.\r\n\r\n<img class=\"size-medium wp-image-1730 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.5-ex.-8-300x130.webp\" alt=\"\" width=\"300\" height=\"130\" \/>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 9: Multiplying Two Matrices<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven <em>A<\/em> and <em>B<\/em>:\r\n\r\n(a) Find <em>AB<\/em>.\r\n\r\n(b) Find <em>BA<\/em>.\r\n\r\n[latex] A=\\begin{bmatrix} -1 &amp; 2 &amp; 3 \\\\ 4 &amp; 0 &amp; 5 \\end{bmatrix} \\quad \\text{and} \\quad B=\\begin{bmatrix} 5 &amp; -1 \\\\ -4 &amp; 0 \\\\ 2 &amp; 3 \\end{bmatrix} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>(a) As the dimensions of <em>A<\/em> are [latex] 2\\times 3 [\/latex] and the dimensions of <em>B<\/em> are [latex] 3\\times 2, [\/latex] these matrices can be multiplied together because the number of columns in <em>A<\/em> matches the number of rows in <em>B<\/em>. The resulting product will be a [latex] 2\\times 2 [\/latex] matrix, the number of rows in <em>A<\/em> by the number of columns in <em>B<\/em>.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} A &amp;=&amp; \\begin{bmatrix} -1 &amp; 2 &amp; 3 \\\\ 4 &amp; 0 &amp; 5 \\end{bmatrix}\\begin{bmatrix} 5 &amp; -1 \\\\ -4 &amp; 0 \\\\ 2 &amp; 3 \\end{bmatrix} \\\\ &amp;=&amp; \\begin{bmatrix} -1(5)+2(-4)+3(2) &amp; -1(-1)+2(0)+3(3) \\\\ 4(5)+0(-4)+5(2) &amp; 4(-1)+0(0)+5(3) \\end{bmatrix} \\\\ &amp;=&amp; \\begin{bmatrix} -7 &amp; 10 \\\\ 30 &amp; 11 \\end{bmatrix} \\end{array} [\/latex]<\/p>\r\n(b) The dimensions of <em>B<\/em> are [latex] 3\\times 2 [\/latex] and the dimensions of <em>A<\/em> are [latex] 2\\times 3 [\/latex] The inner dimensions match so the product is defined and will be a [latex] 3\\times 3 [\/latex] matrix.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll} BA &amp;=&amp; \\begin{bmatrix} 5 &amp; -1 \\\\ -4 &amp; 0 \\\\ 2 &amp; 3 \\end{bmatrix}\\begin{bmatrix} -1 &amp; 2 &amp; 3 \\\\ 4 &amp; 0 &amp; 5 \\end{bmatrix} \\\\ &amp;=&amp; \\begin{bmatrix} 5(-1)+-1(4) &amp; 5(2)+-1(0) &amp; 5(3)+-1(5) \\\\ -4(-1)+0(4) &amp; -4(2)+0(0) &amp; 04(3)+0(5) \\\\ 2(-1)+3(4) &amp; 2(2)+3(0) &amp; 2(3)+3(5) \\end{bmatrix} \\\\ &amp;=&amp; \\begin{bmatrix} -9 &amp; 10 &amp; 10 \\\\ 4 &amp; 08 &amp; -12 \\\\ 10 &amp; 4 &amp; 21 \\end{bmatrix} \\end{array} [\/latex]<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\nNotice that the products <em>AB<\/em> and <em>BA<\/em> are not equal.\r\n<p style=\"text-align: center;\">[latex] AB=\\begin{bmatrix} -7 &amp; 10 \\\\ 30 &amp; 11 \\end{bmatrix}\\not=\\begin{bmatrix} -9 &amp; 10 &amp; 10 \\\\ 4 &amp; -8 &amp; -12 \\\\ 10 &amp; 4 &amp; 21 \\end{bmatrix}=BA [\/latex]<\/p>\r\nThis illustrates the fact that matrix multiplication is not commutative.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Q: Is it possible for <em data-effect=\"italics\">AB<\/em> to be defined but not <em data-effect=\"italics\">BA<\/em>?<\/strong>\r\n\r\n<em>A: Yes, consider a matrix A with dimension and matrix B with dimension For the product AB the inner dimensions are 4 and the product is defined, but for the product BA the inner dimensions are 2 and 3 so the product is undefined.<\/em>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 10: Using Matrices in Real-World Problems<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nLet\u2019s return to the problem presented at the opening of this section. We have Table 3, representing the equipment needs of two soccer teams.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 3<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 33.3333%;\"><\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Wildcats<\/strong><\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Mud Cats<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Goals<\/strong><\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">6<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">10<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Balls<\/strong><\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">30<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">24<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Jerseys<\/strong><\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">14<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">20<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe are also given the prices of the equipment, as shown in Table 4.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 4<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\"><strong>Goal<\/strong><\/td>\r\n<td style=\"width: 50%; text-align: center;\">$300<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\"><strong>Ball<\/strong><\/td>\r\n<td style=\"width: 50%; text-align: center;\">$10<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\"><strong>Jersey<\/strong><\/td>\r\n<td style=\"width: 50%; text-align: center;\">$30<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe will convert the data to matrices. Thus, the equipment need matrix is written as\r\n<p style=\"text-align: center;\">[latex] E=\\begin{bmatrix} 6 &amp; 10 \\\\ 30 &amp; 24 \\\\ 14 &amp; 20 \\end{bmatrix} [\/latex]<\/p>\r\nThe cost matrix is written as\r\n\r\n[latex] C=\\begin{bmatrix} 300 &amp; 10 ^ 30 \\end{bmatrix} [\/latex]\r\n\r\nWe perform matrix multiplication to obtain costs for the equipment.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rll}\u00a0 C &amp;=&amp; \\begin{bmatrix} 300 &amp; 10 &amp; 30 \\end{bmatrix}\\begin{bmatrix} 6 &amp; 10 \\\\ 30 &amp; 24 \\\\ 14 &amp; 20 \\end{bmatrix} \\\\ &amp;=&amp; \\begin{bmatrix} 300(6)+10(30)+30(14) &amp; 300(10)+10(24)+30(20) \\end{bmatrix} \\\\ &amp;=&amp; \\begin{bmatrix} 2,520 &amp; 3,840 \\end{bmatrix} \\end{array} [\/latex]<\/p>\r\nThe total cost for equipment for the Wildcats is $2,520, and the total cost for equipment for the Mud Cats is $3,840.\r\n\r\n<section>\r\n<div><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a matrix operation, evaluate using a calculator.<\/strong>\r\n<ol>\r\n \t<li>Save each matrix as a matrix variable [latex] [A], [B], [C], \\ldots [\/latex]<\/li>\r\n \t<li>Enter the operation into the calculator, calling up each matrix variable as needed.<\/li>\r\n \t<li>If the operation is defined, the calculator will present the solution matrix; if the operation is undefined, it will display an error message.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 11: Using a Calculator to Perform Matrix Operations<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind [latex] AB-C [\/latex] given\r\n<p style=\"text-align: center;\">[latex] A=\\begin{bmatrix} -15 &amp; 25 &amp; 32 \\\\ 41 &amp; -7 &amp; -28 \\\\ 10 &amp; 34 &amp; -2 \\end{bmatrix}, B=\\begin{bmatrix} 45 &amp; 21 &amp; -37 \\\\ -24 &amp; 52 &amp; 19 \\\\ 6 &amp; -48 &amp; -31 \\end{bmatrix}, \\quad \\text{and} \\quad C=\\begin{bmatrix} -100 &amp; -89 &amp; -98 \\\\ 25 &amp; -56 &amp; 74 \\\\ -67 &amp; 42 &amp; -75 \\end{bmatrix}. [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>On the matrix page of the calculator, we enter matrix <em>A<\/em> above as the matrix variable [latex] [A] [\/latex] matrix <em>B<\/em> above as the matrix variable [latex] [B] [\/latex] and matrix <em>C<\/em> above as the matrix variable [latex] [C]\/ [\/latex]\r\n\r\nOn the home screen of the calculator, we type in the problem and call up each matrix variable as needed.\r\n<p style=\"text-align: center;\">[latex] [A][B]-[C] [\/latex]<\/p>\r\nThe calculator gives us the following matrix.\r\n<p style=\"text-align: center;\">[latex] \\begin{bmatrix} -983 &amp; -462 &amp; 136 \\\\ 1,820 &amp; 1,897 &amp; -856 \\\\ -311 &amp; 2,032 &amp; 413 \\end{bmatrix} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Media<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1165137463912\">Access these online resources for additional instruction and practice with matrices and matrix operations.<\/p>\r\n\r\n<ul id=\"eip-id1165137681215\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/matrixdimen\" target=\"_blank\" rel=\"noopener nofollow\">Dimensions of a Matrix<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/matrixaddsub\" target=\"_blank\" rel=\"noopener nofollow\">Matrix Addition and Subtraction<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/matrixoper\" target=\"_blank\" rel=\"noopener nofollow\">Matrix Operations<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/matrixmult\" target=\"_blank\" rel=\"noopener nofollow\">Matrix Multiplication<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\r\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">7.5 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id1165133245136\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1165134301522\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1165134301527\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134301529\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134301527-solution\">1<\/a><span class=\"os-divider\">. <\/span>Can we add any two matrices together? If so, explain why; if not, explain why not and give an example of two matrices that cannot be added together.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134183745\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134183746\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>Can we multiply any column matrix by any row matrix? Explain why or why not.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134183750\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134183752\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134183750-solution\">3<\/a><span class=\"os-divider\">. <\/span>Can both the products <em>AB<\/em> and <em>BA<\/em> be defined? If so, explain how; if not, explain why.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>\r\n<div id=\"fs-id1165137629348\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137629349\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>Can any two matrices of the same size be multiplied? If so, explain why, and if not, explain why not and give an example of two matrices of the same size that cannot be multiplied together.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133088804\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133088805\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133088804-solution\">5<\/a><span class=\"os-divider\">. <\/span>Does matrix multiplication commute? That is, does [latex] AB=BA? [\/latex] If so, prove why it does. If not, explain why it does not.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165137770942\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id1165137770947\">For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.<\/p>\r\n<p style=\"text-align: center;\">[latex] A=\\begin{bmatrix} 1 &amp; 3 \\\\ 0 &amp; 7 \\end{bmatrix}, B= \\begin{bmatrix} 2 &amp; 14 \\\\ 22 &amp; 6 \\end{bmatrix}, C= \\begin{bmatrix} 1 &amp; 5 \\\\ 8 &amp; 92 \\\\ 12 &amp; 6 \\end{bmatrix}, D= \\begin{bmatrix} 10 &amp; 14 \\\\ 7 &amp; 2 \\\\ 5 &amp; 61 \\end{bmatrix}, E= \\begin{bmatrix} 6 &amp; 12 \\\\ 14 &amp; 5 \\end{bmatrix}, F= \\begin{bmatrix} 0 &amp; 9 \\\\ 78 &amp; 17 \\\\ 15 &amp; 4 \\end{bmatrix} [\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165135381338\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134544655\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex] A+B [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165135481913\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135481914\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135481913-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex] C+D [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132035314\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132035315\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex] A+C [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135452967\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135452968\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135452967-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex] B-E [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135388909\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135388910\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex] C+F [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134589497\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134589498\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134589497-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex] D-B [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165133141433\">For the following exercises, use the matrices below to perform scalar multiplication.<\/p>\r\n<p style=\"text-align: center;\">[latex] A=\\begin{bmatrix} 4 &amp; 6 \\\\ 13 &amp; 12 \\end{bmatrix}, B=\\begin{bmatrix} 3 &amp; 9\\\\ 21 &amp; 12 \\\\ 0 &amp; 64 \\end{bmatrix}, C=\\begin{bmatrix} 16 &amp; 3 &amp; 7 &amp; 18 \\\\ 90 &amp; 5 &amp; 3 &amp; 29 \\end{bmatrix}, D=\\begin{bmatrix} 18 &amp; 12 &amp; 13 \\\\ 8 &amp; 14 &amp; 6 \\\\ 7 &amp; 4 &amp; 21 \\end{bmatrix} [\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165135672778\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135672779\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex] 5A [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134556459\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134556460\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134556459-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex] 3B [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132935126\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132935127\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex] -2B [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135367787\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135367788\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135367787-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex] -4C [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135445702\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135445703\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex] \\frac{1}{2}C [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137896991\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137896992\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165137896991-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex] 100D [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135256020\">For the following exercises, use the matrices below to perform matrix multiplication.<\/p>\r\n<p style=\"text-align: center;\">[latex] A=\\begin{bmatrix} -1 &amp; 5 \\\\ 3 &amp; 2 \\end{bmatrix}, B=\\begin{bmatrix} 3 &amp; 6 &amp; 4 \\\\ -8 &amp; 0 &amp; 12 \\end{bmatrix}, C=\\begin{bmatrix} 4 &amp; 10 \\\\ -2 &amp; 6 \\\\ 5 &amp; 9 \\end{bmatrix}, D=\\begin{bmatrix} 2 &amp; -3 &amp; 12 \\\\ 9 &amp; 3 &amp; 1 \\\\ 0 &amp; 8 &amp; -10 \\end{bmatrix} [\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165132332419\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132332420\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex] AB [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133103836\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133103837\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133103836-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex] BC [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135649504\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135649505\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex] CA [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134282176\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134282177\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134282176-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex] BD [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133355960\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137736220\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex] DC [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135367683\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135367684\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135367683-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex] CB [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165134381432\">For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.<\/p>\r\n<p style=\"text-align: center;\">[latex] A=\\begin{bmatrix} 2 &amp; -5 \\\\ 6 &amp;\u00a0 7 \\end{bmatrix}, B=\\begin{bmatrix}\u00a0 -9 &amp; 6 \\\\ -4 &amp; 2 \\end{bmatrix}, C=\\begin{bmatrix} 0 &amp; 9 \\\\ 7 &amp; 1 \\end{bmatrix}, D=\\begin{bmatrix} -8 &amp; 7 &amp; -5 \\\\ 4 &amp; 3 &amp; 2 \\\\ 0 &amp; 9 &amp; 2 \\end{bmatrix}, E=\\begin{bmatrix} 4 &amp; 5 &amp; 3 \\\\ 7 &amp; -6 &amp; -5 \\\\ 1 &amp; 0 &amp; 9 \\end{bmatrix} [\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165134552652\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134552653\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex] A+B-C [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135320576\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135320577\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135320576-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex] 4A+5D [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137683016\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137683017\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex] 2C+B [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134042234\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134267898\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134042234-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex] 3D+4E [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133176699\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133176700\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex] C-0.5D [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134042465\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134042466\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134042465-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex] 100D-10E [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135196589\">For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: [latex] A^2=A\\cdot A [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] A=\\begin{bmatrix} -10 &amp; 20 \\\\ 5 &amp; 25 \\end{bmatrix}, B=\\begin{bmatrix}\u00a0 40 &amp; 10 \\\\ -20 &amp; 30 \\end{bmatrix}, C=\\begin{bmatrix} -1 &amp; 0\\\\ 0 &amp; -1 \\\\ 1 &amp; 0 \\end{bmatrix} [\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165135518214\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135518215\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">. [latex] AB [\/latex]<\/span>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135332915\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135332916\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135332915-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex] BA [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165131958337\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165131958338\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex] CA [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137809962\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137809963\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165137809962-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex] BC [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135662475\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134112957\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex] A^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135440347\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135440348\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135440347-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex] B^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133157399\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133157400\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex] C^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133214671\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133214672\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133214671-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex] B^2A^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133324890\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133324891\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex] A^2B^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165131891720\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165131891721\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165131891720-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex] (AB)^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135595090\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135595091\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex] (BA)^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165131949634\">For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: [latex] A^2=A\\cdot A [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] A=\\begin{bmatrix} 1 &amp; 0 \\\\ 2 &amp; 3 \\end{bmatrix}, B=\\begin{bmatrix} -2 &amp; 3 &amp; 4 \\\\ -1 &amp; 1 &amp; -5 \\end{bmatrix}, C=\\begin{bmatrix} 0.5 &amp; 0.1 \\\\ 1 &amp; 0.2 \\\\ -0.5 &amp; 0.3 \\end{bmatrix}, D=\\begin{bmatrix} 1 &amp; 0 &amp; -1 \\\\ -6 &amp; 7 &amp; 5 \\\\ 4 &amp; 2 &amp; 1 \\end{bmatrix} [\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165135177605\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135177606\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135177605-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex] AB [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133092748\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133092749\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex] BA [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133134811\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133134812\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133134811-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex] BD [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135701628\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135701629\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex] DC [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135320241\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135320242\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135320241-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex] D^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135354776\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135354777\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex] A^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135421447\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135421448\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135421447-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex] D^3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135702617\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135702618\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex] (AB)C [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135154488\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135154489\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135154488-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex] A(BC) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165137900018\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Technology<\/h3>\r\n<p id=\"fs-id1165135455953\">For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution.<\/p>\r\n<p style=\"text-align: center;\">[latex] A=\\begin{bmatrix} -2 &amp; 0 &amp; 9 \\\\ 1 &amp; 8 &amp; -3 \\\\ 0.5 &amp; 4 &amp; 5 \\end{bmatrix}, B=\\begin{bmatrix} 0.5 &amp; 3 &amp; 0 \\\\ -4 &amp; 1 &amp; 6 \\\\ 8 &amp; 7 &amp; 2 \\end{bmatrix}, C=\\begin{bmatrix} 1 &amp; 0 &amp; 1 \\\\ 0 &amp; 1 &amp; 0\\\\ 1 &amp; 0 &amp; 1 \\end{bmatrix} [\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165135369165\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135369166\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex] AB [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165135546873\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135546874\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135546873-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex] BA [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134407090\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134407091\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex] CA [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134038135\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134038136\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134038135-solution\">53<\/a><span class=\"os-divider\">. <\/span> [latex] BC [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132986100\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132986101\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span> [latex] ABC [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165134188873\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extensions<\/h3>\r\n<p id=\"fs-id1165137407686\">For the following exercises, use the matrix below to perform the indicated operation on the given matrix.<\/p>\r\n<p style=\"text-align: center;\">[latex] B=\\begin{bmatrix} 1 &amp; 0 &amp; 0 \\\\ 0 &amp; 0 &amp; 1 \\\\ 0 &amp; 1 &amp; 0 \\end{bmatrix} [\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165135298429\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135298430\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135298429-solution\">55<\/a><span class=\"os-divider\">. <\/span> [latex] B^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165133088779\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133088780\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span> [latex] B^3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133377041\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133377042\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133377041-solution\">57<\/a><span class=\"os-divider\">. <\/span> [latex] B^4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134225694\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134225695\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span> [latex] B^5 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135455009\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135455010\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135455009-solution\">59<\/a><span class=\"os-divider\">. <\/span>Using the above questions, find a formula for [latex] B^n. [\/latex] Test the formula for [latex] B^{201} [\/latex] and [latex] B^{202}, [\/latex] using a calculator.\r\n\r\n<\/div>\r\n<\/section><\/div>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_508a4d4e-0136-4c6c-87b8-e210022d69b4\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>In this section, you will:<\/p>\n<ul>\n<li>Find the sum and difference of two matrices.<\/li>\n<li>Find scalar multiples of a matrix.<\/li>\n<li>Find the product of two matrices.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<figure id=\"attachment_1723\" aria-describedby=\"caption-attachment-1723\" style=\"width: 297px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1723\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.5-fig-1-297x300.webp\" alt=\"\" width=\"297\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.5-fig-1-297x300.webp 297w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.5-fig-1-65x66.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.5-fig-1-225x227.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.5-fig-1-350x353.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.5-fig-1.webp 651w\" sizes=\"auto, (max-width: 297px) 100vw, 297px\" \/><figcaption id=\"caption-attachment-1723\" class=\"wp-caption-text\">Figure 1 (credit: &#8220;SD Dirk,&#8221; Flickr)<\/figcaption><\/figure>\n<p id=\"fs-id1165137862059\">Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season. Table 1 shows the needs of both teams.<\/p>\n<div id=\"Table_09_05_01\" class=\"os-table\">\n<table class=\"grid\" style=\"height: 96px; width: 356px;\" data-id=\"Table_09_05_01\">\n<caption>Table 1<\/caption>\n<thead>\n<tr style=\"height: 24px;\">\n<th style=\"height: 24px; width: 107.733px;\" scope=\"col\"><\/th>\n<th style=\"height: 24px; width: 122.233px;\" scope=\"col\" data-align=\"center\">Wildcats<\/th>\n<th style=\"height: 24px; width: 104.033px;\" scope=\"col\" data-align=\"center\">Mud Cats<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 24px;\">\n<td style=\"height: 24px; width: 107.733px;\" data-align=\"center\"><strong>Goals<\/strong><\/td>\n<td style=\"height: 24px; width: 122.233px;\" data-align=\"center\">6<\/td>\n<td style=\"height: 24px; width: 104.033px;\" data-align=\"center\">10<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"height: 24px; width: 107.733px;\" data-align=\"center\"><strong>Balls<\/strong><\/td>\n<td style=\"height: 24px; width: 122.233px;\" data-align=\"center\">30<\/td>\n<td style=\"height: 24px; width: 104.033px;\" data-align=\"center\">24<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"height: 24px; width: 107.733px;\" data-align=\"center\"><strong>Jerseys<\/strong><\/td>\n<td style=\"height: 24px; width: 122.233px;\" data-align=\"center\">14<\/td>\n<td style=\"height: 24px; width: 104.033px;\" data-align=\"center\">20<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\">A goal costs $300; a ball costs $10; and a jersey costs $30. How can we find the total cost for the equipment needed for each team? In this section, we discover a method in which the data in the soccer equipment table can be displayed and used for calculating other information. Then, we will be able to calculate the cost of the equipment.<\/div>\n<\/div>\n<section id=\"fs-id1165132962023\" data-depth=\"1\">\n<h2 data-type=\"title\">Finding the Sum and Difference of Two Matrices<\/h2>\n<p id=\"fs-id1165134430367\">To solve a problem like the one described for the soccer teams, we can use a <span id=\"term-00005\" class=\"no-emphasis\" data-type=\"term\">matrix<\/span>, which is a rectangular array of numbers. A <span id=\"term-00006\" class=\"no-emphasis\" data-type=\"term\">row<\/span> in a matrix is a set of numbers that are aligned horizontally. A <span id=\"term-00007\" class=\"no-emphasis\" data-type=\"term\">column<\/span> in a matrix is a set of numbers that are aligned vertically. Each number is an <span id=\"term-00008\" class=\"no-emphasis\" data-type=\"term\">entry<\/span>, sometimes called an element, of the matrix. Matrices (plural) are enclosed in [ ] or ( ), and are usually named with capital letters. For example, three matrices named\u00a0<em>A, B<\/em>\u00a0and\u00a0<em>C<\/em> are shown below.<\/p>\n<p style=\"text-align: center;\">[latex]A = \\begin{bmatrix} 1 & 2 \\\\ 3 & 4 \\end{bmatrix}, B = \\begin{bmatrix} 1 & 2 & 7 \\\\ 0 & -5 & 6 \\\\ 7 & 8 & 2 \\end{bmatrix}, C = \\begin{bmatrix} -1 & 3 \\\\ 0 & 2 \\\\ 3 & 1 \\end{bmatrix}[\/latex]<\/p>\n<section id=\"fs-id1165132938226\" data-depth=\"2\">\n<h3 data-type=\"title\">Describing Matrices<\/h3>\n<p id=\"fs-id1165134060293\">A matrix is often referred to by its size or dimensions: [latex]m\\times n[\/latex] indicating <em>m<\/em> rows and <em>n<\/em> columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix <em>A<\/em> identified as [latex]a_{ij},[\/latex] we look for the entry in row <em>i<\/em>, column <em>j<\/em>. In matrix <em>A<\/em>, shown below, the entry in row 2, column 3 is [latex]a_{23}.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]A = \\begin{bmatrix} a_{11} & a_{12} & a_{13} \\\\ a_{21} & a_{22} & a_{23} \\\\ a_{31} & a_{32} & a_{33} \\end{bmatrix}[\/latex]<\/p>\n<p id=\"fs-id1165137629048\">A square matrix is a matrix with dimensions [latex]n\\times n[\/latex] meaning that it has the same number of rows as columns. The [latex]3\\times 3[\/latex] matrix above is an example of a square matrix.<\/p>\n<p id=\"fs-id1165137400185\">A row matrix is a matrix consisting of one row with dimensions [latex]1\\times n.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{bmatrix} a_{11} & a_{12} & a_{13} \\end{bmatrix}[\/latex]<\/p>\n<p id=\"fs-id1165137475278\">A column matrix is a matrix consisting of one column with dimensions [latex]m\\times 1.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{bmatrix} a_{11} \\\\ a_{21} \\\\ a_{31} \\end{bmatrix}[\/latex]<\/p>\n<p id=\"fs-id1165135307784\">A matrix may be used to represent a system of equations. In these cases, the numbers represent the coefficients of the variables in the system. Matrices often make solving systems of equations easier because they are not encumbered with variables. We will investigate this idea further in the next section, but first we will look at basic matrix operations.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Matrices<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A <strong>matrix <\/strong>is a rectangular array of numbers that is usually named by a capital letter:\u00a0<em>A, B, C <\/em>and so on. Each entry in a matrix is referred to as [latex]a_{ij},[\/latex] such that <em>i<\/em> represents the row and\u00a0<em>j<\/em> represents the column. Matrices are often referred to by their dimensions: [latex]m\\times n[\/latex] indicating <em>m\u00a0<\/em>rows and\u00a0<em>n\u00a0<\/em>columns.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 1: Finding the Dimensions of the Given Matrix and Locating Entries<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given matrix\u00a0<em>A:<\/em><\/p>\n<p>(a) What are the dimensions of matrix\u00a0<em>A?<\/em><\/p>\n<p>(b) What are the entries at [latex]a_{31}[\/latex] and [latex]a_{22}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]A=\\begin{bmatrix} 2 & 1 & 0 \\\\ 2 & 4 & 7 \\\\ 3 & 1 & -2 \\end{bmatrix}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>(a) The dimensions are [latex]3\\times 3[\/latex] because there are three rows and three columns.<\/p>\n<p>(b) Entry [latex]a_{31}[\/latex] is the number at row 3, column 1, which is 3. The entry [latex]a_{22}[\/latex] is the number at row 2, column 2, which is 4. Remember, the row comes first, then the column.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<section id=\"fs-id1165135499659\" data-depth=\"2\">\n<h3 data-type=\"title\">Adding and Subtracting Matrices<\/h3>\n<p id=\"fs-id1165137602057\">We use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries.<\/p>\n<p id=\"fs-id1165137590185\">In order to do this, the entries must correspond. Therefore, <em data-effect=\"italics\">addition and subtraction of matrices is only possible when the matrices have the same dimensions<\/em>. We can add or subtract a [latex]3\\times 3[\/latex] matrix and another [latex]3\\times 3[\/latex] matrix, but we cannot add or subtract a [latex]2\\times 3[\/latex] matrix and a [latex]3\\times 3[\/latex] matrix because some entries in one matrix will not have a corresponding entry in the other matrix.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Adding and Subtracting Matrices<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given matrices\u00a0<em>A<\/em> and\u00a0<em>B<\/em> of like dimensions, addition and subtraction of\u00a0<em>A<\/em> and\u00a0<em>B\u00a0<\/em>will produce matrix\u00a0<em>C\u00a0<\/em>or matrix\u00a0<em>D <\/em>of the same dimension.<\/p>\n<p style=\"text-align: center;\">[latex]A+B=C[\/latex] such that [latex]a_{ij}+b_{ij}=c_{ij}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]A-B=D[\/latex] such that [latex]a_{ij}-b_{ij}=d_{ij}[\/latex]<\/p>\n<p>Matrix addition is commutative.<\/p>\n<p style=\"text-align: center;\">[latex]A+B=B+A[\/latex]<\/p>\n<p>It is also associative.<\/p>\n<p style=\"text-align: center;\">[latex](A+B)+C=A+(B+C)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 2: Finding the Sum of Matrices<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the sum of <em>A<\/em> and <em>B<\/em>, given<\/p>\n<p style=\"text-align: center;\">[latex]A = \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\quad \\text{and} \\quad B = \\begin{bmatrix} e & f \\\\ g & h \\end{bmatrix}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Add corresponding entries.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} A+B &=& \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix}+\\begin{bmatrix} e & f \\\\ g & h \\end{bmatrix} \\\\ \\\\ &=& \\begin{bmatrix} a+e & b+f \\\\ c+g & d+h \\end{bmatrix} \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3: Adding Matrix <em>A\u00a0<\/em>and Matrix\u00a0<em>B<\/em><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the sum of <em>A<\/em> and <em>B<\/em>.<\/p>\n<p>[latex]A = \\begin{bmatrix} 4 & 1 \\\\ 3 & 2 \\end{bmatrix} \\quad \\text{and} \\quad B = \\begin{bmatrix} 5 & 9 \\\\ 0 & 7 \\end{bmatrix}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Add corresponding entries. Add the entry in row 1, column 1, [latex]a_{11},[\/latex] of matrix <em>A<\/em> to the entry in row 1, column 1, [latex]b_{11},[\/latex] of <em>B<\/em>. Continue the pattern until all entries have been added.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} A+B &=& \\begin{bmatrix} 4 & 1 \\\\ 3 & 2 \\end{bmatrix}+\\begin{bmatrix} 5 & 9 \\\\ 0 & 7 \\end{bmatrix} \\\\ \\\\ &=& \\begin{bmatrix} 4+5 & 1+9 \\\\ 3+0 & 2+7 \\end{bmatrix} \\\\ \\\\ &=& \\begin{bmatrix} 9 & 10 \\\\ 3 & 9 \\end{bmatrix} \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 4: Finding the Difference of Two Matrices<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the difference of <em>A<\/em> and <em>B<\/em>.<\/p>\n<p>[latex]A = \\begin{bmatrix} -2 & 3 \\\\ 0 & 1 \\end{bmatrix} \\quad \\text{and} \\quad B = \\begin{bmatrix} 8 & 1 \\\\ 5 & 4 \\end{bmatrix}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>We subtract the corresponding entries of each matrix.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} A+B &=& \\begin{bmatrix} -2 & 3 \\\\ 0 & 1 \\end{bmatrix}-\\begin{bmatrix} 8 & 1 \\\\ 5 & 4 \\end{bmatrix} \\\\ \\\\ &=& \\begin{bmatrix} -2-8 & 3-1 \\\\ 0-5 & 1-4 \\end{bmatrix} \\\\ \\\\ &=& \\begin{bmatrix} -10 & 2 \\\\ -5 & -3 \\end{bmatrix} \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 5: Finding the Sum and Difference of Two 3&#215;3 Matrices<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given <em>A<\/em> and <em>B<\/em>:<\/p>\n<p>(a) Find the sum.<\/p>\n<p>(b) Find the difference.<\/p>\n<p>[latex]A = \\begin{bmatrix} 2 & -10 & -2 \\\\ 14 & 12 & 10 \\end{bmatrix} \\quad \\text{and} \\quad B = \\begin{bmatrix} 6 & 10 & -2 \\\\ 0 & =12 & -4 \\\\ -5 & 2 & -2\\end{bmatrix}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>(a) Add the corresponding entries.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} A+B &=& \\begin{bmatrix} 2 & -10 & -2 \\\\ 14 & 12 & 10 \\\\ 4 & -2 & 2 \\end{bmatrix}+\\begin{bmatrix} 6 & 10 & -2 \\\\ 0 & -12 & -4 \\\\ -5 & 2 & -2 \\end{bmatrix} \\\\ \\\\ &=& \\begin{bmatrix} 2+6 & -10+1- & -2-2 \\\\ 14+0 & 12-12 & 10-4 \\\\ 4-5 & -2+2 & 2-2 \\end{bmatrix} \\\\ \\\\ &=& \\begin{bmatrix} 8 & 0 & -4 \\\\ 14 & 0 & 6 \\\\ -1 & 0 & 0 \\end{bmatrix} \\end{array}[\/latex]<\/p>\n<p>(b) Subtract the corresponding entries.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} A+B &=& \\begin{bmatrix} 2 & -10 & -2 \\\\ 14 & 12 & 10 \\\\ 4 & -2 & 2 \\end{bmatrix}-\\begin{bmatrix} 6 & 10 & -2 \\\\ 0 & -12 & -4 \\\\ -5 & 2 & -2 \\end{bmatrix} \\\\ \\\\ &=& \\begin{bmatrix} 2-6 & -10-10 & -2+2 \\\\ 14-0 & 12+12 & 10+4 \\\\ 4+5 & -2-2 & 2+2 \\end{bmatrix} \\\\ \\\\ &=& \\begin{bmatrix} -4 & -20 & 0 \\\\ 14 & 24 & 14 \\\\ 9 & -4 & 4 \\end{bmatrix} \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Add matrix <em>A<\/em> and matrix <em>B<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]A = \\begin{bmatrix} 2 & 6 \\\\ 1 & 0 \\\\ 1 & -3 \\end{bmatrix} \\quad \\text{and} \\quad B = \\begin{bmatrix} 3 & -2 \\\\ 1 & 5 \\\\ -4 & 3 \\end{bmatrix}[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1165134389759\" data-depth=\"1\">\n<h2 data-type=\"title\">Finding Scalar Multiples of a Matrix<\/h2>\n<p id=\"fs-id1165137517260\">Besides adding and subtracting whole matrices, there are many situations in which we need to multiply a matrix by a constant called a scalar. Recall that a <span id=\"term-00014\" class=\"no-emphasis\" data-type=\"term\">scalar<\/span> is a real number quantity that has magnitude, but not direction. For example, time, temperature, and distance are scalar quantities. The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A <strong>scalar multiple<\/strong> is any entry of a matrix that results from scalar multiplication.<\/p>\n<p id=\"fs-id1165137686653\">Consider a real-world scenario in which a university needs to add to its inventory of computers, computer tables, and chairs in two of the campus labs due to increased enrollment. They estimate that 15% more equipment is needed in both labs. The school\u2019s current inventory is displayed in Table 2.<\/p>\n<div id=\"Table_09_05_02\" class=\"os-table\">\n<table class=\"grid\" style=\"height: 96px; width: 384px;\" data-id=\"Table_09_05_02\">\n<caption>Table 2<\/caption>\n<thead>\n<tr style=\"height: 24px;\">\n<th style=\"height: 24px; width: 169.733px;\" scope=\"col\"><\/th>\n<th style=\"height: 24px; width: 93.4167px;\" scope=\"col\" data-align=\"center\">Lab A<\/th>\n<th style=\"height: 24px; width: 98.85px;\" scope=\"col\" data-align=\"center\">Lab B<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 24px;\">\n<td style=\"height: 24px; width: 169.733px;\" data-align=\"center\"><strong>Computers<\/strong><\/td>\n<td style=\"height: 24px; width: 93.4167px;\" data-align=\"center\">15<\/td>\n<td style=\"height: 24px; width: 98.85px;\" data-align=\"center\">27<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"height: 24px; width: 169.733px;\" data-align=\"center\"><strong>Computer Tables<\/strong><\/td>\n<td style=\"height: 24px; width: 93.4167px;\" data-align=\"center\">16<\/td>\n<td style=\"height: 24px; width: 98.85px;\" data-align=\"center\">34<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"height: 24px; width: 169.733px;\" data-align=\"center\"><strong>Chairs<\/strong><\/td>\n<td style=\"height: 24px; width: 93.4167px;\" data-align=\"center\">16<\/td>\n<td style=\"height: 24px; width: 98.85px;\" data-align=\"center\">34<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\">\n<p>Converting the data to a matrix, we have<\/p>\n<p style=\"text-align: center;\">[latex]C_{2013}=\\begin{bmatrix} 15 & 27 \\\\ 16 & 34 \\\\ 16 & 34 \\end{bmatrix}[\/latex]<\/p>\n<p id=\"fs-id1165134583398\">To calculate how much computer equipment will be needed, we multiply all entries in matrix <em>C<\/em> by 0.15.<\/p>\n<p style=\"text-align: center;\">[latex](0.15)C_{2013}=\\begin{bmatrix} (0.15)15 & (0.15)27 \\\\ (0.15)16 & (0.15)34 \\\\ (0.15)16 & (0.15)34 \\end{bmatrix}=\\begin{bmatrix} 2.25 & 4.05 \\\\ 2.4 & 5.1 \\\\ 2.4 & 5.1 \\end{bmatrix}[\/latex]<\/p>\n<p id=\"fs-id1165134259236\">We must round up to the next integer, so the amount of new equipment needed is<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{bmatrix} 3 & 5 \\\\ 3 & 6 \\\\ 3 & 6 \\end{bmatrix}[\/latex]<\/p>\n<p id=\"fs-id1165135169236\">Adding the two matrices as shown below, we see the new inventory amounts.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{bmatrix} 15 & 27 \\\\ 16 & 34 \\\\ 16 & 34 \\end{bmatrix}+\\begin{bmatrix} 3 & 5 \\\\ 3 & 6 \\\\ 3 & 6 \\end{bmatrix}=\\begin{bmatrix} 18 & 32 \\\\ 19 & 40 \\\\ 19 & 40 \\end{bmatrix}[\/latex]<\/p>\n<p id=\"fs-id1165135487154\">This means<\/p>\n<p style=\"text-align: center;\">[latex]C_{2014}=\\begin{bmatrix} 18 & 32 \\\\ 19 & 40 \\\\ 19 & 40 \\end{bmatrix}[\/latex]<\/p>\n<p id=\"fs-id1165135186654\">Thus, Lab A will have 18 computers, 19 computer tables, and 19 chairs; Lab B will have 32 computers, 40 computer tables, and 40 chairs.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Scalar Multiplication<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Scalar multiplication involves finding the product of a constant by each entry in the matrix. Given<\/p>\n<p style=\"text-align: center;\">[latex]A=\\begin{bmatrix} a_{11} & a_{12} \\\\ a_{21} & a_{22}\u00a0 \\end{bmatrix}[\/latex]<\/p>\n<p>the scalar multiple [latex]cA[\/latex] is<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl} cA=c\\begin{bmatrix} a_{11} & a_{12} \\\\ a_{21} & a_{22} \\end{bmatrix} \\\\ &=& \\begin{bmatrix} ca_{11} & ca_{12} \\\\ ca_{21} & ca_{22} \\end{bmatrix} \\end{array}[\/latex]<\/p>\n<p>Scalar multiplication is distributive. For the matrices <em>A<\/em>, <em>B<\/em>, and <em>C<\/em> with scalars <em>a<\/em> and <em>b<\/em>,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl} a(A+B) &=& aA+aB \\\\ (a=b)A &=& aA+bA \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 6: Multiplying the Matrix by a Scalar<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Multiply matrix <em>A<\/em> by the scalar 3.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\begin{bmatrix} 8 & 1 \\\\ 5 & 4 \\end{bmatrix}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Multiply each entry in <em>A<\/em> by the scalar 3.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl} 3A &=& 3\\begin{bmatrix} 8 & 1 \\\\ 5 & 4 \\end{bmatrix} \\\\ &=& \\begin{bmatrix} 3\\cdot 8 & 3\\cdot 1 \\\\ 3\\cdot 5 & 3\\cdot 4\u00a0 \\end{bmatrix} \\\\ &=& \\begin{bmatrix} 24 & 3 \\\\ 15 & 12\u00a0 \\end{bmatrix} \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given matrix <em>B<\/em> find <em>-2B<\/em> where<\/p>\n<p style=\"text-align: center;\">[latex]B=\\begin{bmatrix} 4 & 1 \\\\ 3 & 2\u00a0 \\end{bmatrix}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 7: Finding the Sum of Scalar Multiples<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the sum [latex]3A+2B[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]A= \\begin{bmatrix} 1 & -2 & 0 \\\\ 0 & -1 & 2 \\\\ 4 & 3 & -6\u00a0 \\end{bmatrix}\u00a0 \\quad \\text{and} \\quad B= \\begin{bmatrix}\u00a0 -1 & 2 & 1 \\\\ 0 & -3 & 2 \\\\ 0 & 1 & -4 \\end{bmatrix}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>First, find <em>3A<\/em>, then <em>2B<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl} 3A &=& \\begin{bmatrix} 3\\cdot 1 & 3(-2) & 3\\cdot 0 \\\\ 3\\cdot 0 & 3(-1) & 3\\cdot 2 \\\\ 3\\cdot 4 & 3\\cdot 3 & 3(-6)\u00a0 \\end{bmatrix} \\\\ &=& \\begin{bmatrix} 3 & -6 & 0 \\\\ 0 & -3 & 6 \\\\ 12 & 9 & 18\u00a0 \\end{bmatrix} \\\\ \\\\ 2B &=&\u00a0 \\begin{bmatrix} 2(-1) & 2\\cdot 2 & 2\\cdot 1 \\\\ 2\\cdot 0 & 2(-3) & 2\\cdot 2 \\\\ 2\\cdot 0 & 2\\cdot 1 & 2(-4)\u00a0 \\end{bmatrix} \\\\\u00a0 &=& \\begin{bmatrix} -2 & 4 & 2 \\\\ 0 & -6 & 4 \\\\ 0 & 2 & -8\u00a0 \\end{bmatrix} \\end{array}[\/latex]<\/p>\n<p>Now, add [latex]3A+2B[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl} 3A+2B &=&\u00a0 \\begin{bmatrix} 3 & -6 & 0 \\\\ 0 & -3 & 6 \\\\ 12 & 9 & -18\u00a0 \\end{bmatrix}+ \\begin{bmatrix} -2 & 4 & 2 \\\\ 0 & -6 & 4 \\\\ 0 & 2 & -8\u00a0 \\end{bmatrix} \\\\ &=& \\begin{bmatrix} 3-2 & -6+4 & 0+2 \\\\ 0+0 & -3-6 & 6+4 \\\\ 12+0 & 9+2 & -18-8 \\end{bmatrix} \\\\ &=&\u00a0 \\begin{bmatrix} 1 & -2 & 2 \\\\ 0 & -9 & 10 \\\\ 12 & 11 & -26\u00a0 \\end{bmatrix}\u00a0 \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165133073877\" data-depth=\"1\">\n<h2 data-type=\"title\">Finding the Product of Two Matrices<\/h2>\n<p id=\"fs-id1165134371143\">In addition to multiplying a matrix by a scalar, we can multiply two matrices. Finding the product of two matricesis only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. If <em>A<\/em> is an [latex]m\\times r[\/latex] matrix and <em>B<\/em> is an [latex]r\\times n[\/latex] matrix, then the product matrix <em>AB<\/em> is an [latex]m\\times n[\/latex] matrix. For example, the product <em>AB<\/em> is possible because the number of columns in <em>A<\/em> is the same as the number of rows in <em>B<\/em> If the inner dimensions do not match, the product is not defined.<\/p>\n<p><span id=\"eip-id1165134546098\" data-type=\"media\" data-alt=\"\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1728 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.5-matrix.webp\" alt=\"\" width=\"154\" height=\"93\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.5-matrix.webp 154w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.5-matrix-65x39.webp 65w\" sizes=\"auto, (max-width: 154px) 100vw, 154px\" \/><\/span><\/p>\n<p id=\"fs-id1165135649448\">We multiply entries of <em>A<\/em> with entries of <em>B<\/em> according to a specific pattern as outlined below. The process of matrix multiplication becomes clearer when working a problem with real numbers.<\/p>\n<p id=\"fs-id1165131956653\">To obtain the entries in row <em>i<\/em> of <em>AB<\/em>, we multiply the entries in row <em>i<\/em> of <em>A<\/em> by column <em>j<\/em> in <em>B<\/em> and add. For example, given matrices <em>A<\/em> and <em>B<\/em> where the dimensions of <em>A<\/em> are [latex]2\\times 3[\/latex] and the dimensions of <em>B<\/em> are [latex]3\\times 3[\/latex] the product of <em>AB<\/em> will be a [latex]2\\times 3[\/latex] matrix.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\begin{bmatrix} a_{11} & a_{12} & a_{13} \\\\ a _{21} & a_{22} & a_{23} \\end{bmatrix} \\quad \\text{and} \\quad B=\\begin{bmatrix} b_{11} & b_{12} & b_{13} \\\\ b_{21} & b_{22} & b_{23} \\\\ b_{31} & b_{32} & b_{33} \\end{bmatrix}[\/latex]<\/p>\n<p id=\"fs-id1165134279285\">Multiply and add as follows to obtain the first entry of the product matrix <em>AB<\/em><\/p>\n<ol id=\"fs-id1165134547418\" type=\"1\">\n<li>To obtain the entry in row 1, column 1 of <em>AB<\/em>, multiply the first row in <em>A<\/em> by the first column in <em>B<\/em> and add.<br \/>\n[latex]\\begin{bmatrix} a_{11} & a_{12} & a_{13} \\end{bmatrix} \\begin{bmatrix} b_{11} \\\\ b_{21} \\\\ b_{31} \\end{bmatrix}=a_{11}\\cdot b_{11} + a_{12}\\cdot b_{21} + a_{13}\\cdot b_{31}[\/latex]<\/li>\n<li>To obtain the entry in row 1, column 2 of <em>AB<\/em>, multiply the first row of <em>A<\/em> by the second column in <em>B<\/em> and add.<br \/>\n[latex]\\begin{bmatrix} a_{11} & a_{12} & a_{13} \\end{bmatrix}\\begin{bmatrix} b_{12} \\\\ b_{22} \\\\ b_{32} \\end{bmatrix}=a_{11}\\cdot b_{12} + a_{12}\\cdot b_{22} + a_{13}\\cdot b_{32}[\/latex]<\/li>\n<li>To obtain the entry in row 1, column 3 of <em>AB<\/em>, multiply the first row of <em>A<\/em> by the third column in <em>B<\/em> and add.<br \/>\n[latex]\\begin{bmatrix} a_{11} & a_{12} & a_{13} \\end{bmatrix}\\begin{bmatrix} b_{13} \\\\ b_{23} \\\\ b_{33} \\end{bmatrix}=a_{11}\\cdot b_{13} + a_{12}\\cdot b_{23} + a_{13}\\cdot b_{33}[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1165134589491\">We proceed the same way to obtain the second row of <em>AB<\/em>. In other words, row 2 of <em>A<\/em> times column 1 of <em>B<\/em>; row 2 of <em>A<\/em> times column 2 of <em>B<\/em>; row 2 of <em>A<\/em> times column 3 <em>of<\/em> B. When complete, the product matrix will be<\/p>\n<p>[latex]AB=\\begin{bmatrix} a_{11}\\cdot b_{11} + a_{12}\\cdot b_{21} + a_{13}\\cdot b_{31} & a_{11}\\cdot b_{12} + a_{12}\\cdot b_{22} + a_{13}\\cdot b_{32} & a_{11}\\cdot b_{13} + a_{12}\\cdot b_{23} + a_{13}\\cdot b_{33} \\\\ a_{21}\\cdot b_{11} + a_{22}\\cdot b_{21} + a_{23}\\cdot b_{31} & a_{21}\\cdot b_{12} + a_{22}\\cdot b_{22} + a_{23}\\cdot b_{32} & a_{21}\\cdot b_{13} + a_{22}\\cdot b_{23} + a_{23}\\cdot b_{33} \\end{bmatrix}[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Properties of Matrix Multiplication<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>For the matrices <em>A<\/em>, <em>B<\/em>, and <em>C<\/em> the following properties hold.<\/p>\n<ul>\n<li>Matrix multiplication is associative: [latex](AB)C=A(BC).[\/latex]<\/li>\n<li>Matrix multiplication is distributive: [latex]\\begin{array}{rl} C(A+B) &=& CA+CB, \\\\ (A+B)C &=& AC+BC. \\end{array}[\/latex]<\/li>\n<\/ul>\n<p>Note that matrix multiplication is not commutative.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 8: Multiplying Two Matrices<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Multiply matrix <em>A<\/em> and matrix <em>B<\/em>.<\/p>\n<p>[latex]A=\\begin{bmatrix} 1 & 2 \\\\ 3 & 4 \\end{bmatrix} \\quad \\text{and} \\quad B=\\begin{bmatrix} 5 & 6 \\\\ 7 & 8 \\end{bmatrix}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>First, we check the dimensions of the matrices. Matrix <em>A<\/em> has dimensions [latex]2\\times 2[\/latex] and matrix <em>B<\/em> has dimensions [latex]2\\times 2[\/latex] The inner dimensions are the same so we can perform the multiplication. The product will have the dimensions [latex]2\\times 2.[\/latex]<\/p>\n<p>We perform the operations outlined previously.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1730 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.5-ex.-8-300x130.webp\" alt=\"\" width=\"300\" height=\"130\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.5-ex.-8-300x130.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.5-ex.-8-65x28.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.5-ex.-8-225x97.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.5-ex.-8-350x152.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/7.5-ex.-8.webp 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 9: Multiplying Two Matrices<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given <em>A<\/em> and <em>B<\/em>:<\/p>\n<p>(a) Find <em>AB<\/em>.<\/p>\n<p>(b) Find <em>BA<\/em>.<\/p>\n<p>[latex]A=\\begin{bmatrix} -1 & 2 & 3 \\\\ 4 & 0 & 5 \\end{bmatrix} \\quad \\text{and} \\quad B=\\begin{bmatrix} 5 & -1 \\\\ -4 & 0 \\\\ 2 & 3 \\end{bmatrix}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>(a) As the dimensions of <em>A<\/em> are [latex]2\\times 3[\/latex] and the dimensions of <em>B<\/em> are [latex]3\\times 2,[\/latex] these matrices can be multiplied together because the number of columns in <em>A<\/em> matches the number of rows in <em>B<\/em>. The resulting product will be a [latex]2\\times 2[\/latex] matrix, the number of rows in <em>A<\/em> by the number of columns in <em>B<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} A &=& \\begin{bmatrix} -1 & 2 & 3 \\\\ 4 & 0 & 5 \\end{bmatrix}\\begin{bmatrix} 5 & -1 \\\\ -4 & 0 \\\\ 2 & 3 \\end{bmatrix} \\\\ &=& \\begin{bmatrix} -1(5)+2(-4)+3(2) & -1(-1)+2(0)+3(3) \\\\ 4(5)+0(-4)+5(2) & 4(-1)+0(0)+5(3) \\end{bmatrix} \\\\ &=& \\begin{bmatrix} -7 & 10 \\\\ 30 & 11 \\end{bmatrix} \\end{array}[\/latex]<\/p>\n<p>(b) The dimensions of <em>B<\/em> are [latex]3\\times 2[\/latex] and the dimensions of <em>A<\/em> are [latex]2\\times 3[\/latex] The inner dimensions match so the product is defined and will be a [latex]3\\times 3[\/latex] matrix.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll} BA &=& \\begin{bmatrix} 5 & -1 \\\\ -4 & 0 \\\\ 2 & 3 \\end{bmatrix}\\begin{bmatrix} -1 & 2 & 3 \\\\ 4 & 0 & 5 \\end{bmatrix} \\\\ &=& \\begin{bmatrix} 5(-1)+-1(4) & 5(2)+-1(0) & 5(3)+-1(5) \\\\ -4(-1)+0(4) & -4(2)+0(0) & 04(3)+0(5) \\\\ 2(-1)+3(4) & 2(2)+3(0) & 2(3)+3(5) \\end{bmatrix} \\\\ &=& \\begin{bmatrix} -9 & 10 & 10 \\\\ 4 & 08 & -12 \\\\ 10 & 4 & 21 \\end{bmatrix} \\end{array}[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>Notice that the products <em>AB<\/em> and <em>BA<\/em> are not equal.<\/p>\n<p style=\"text-align: center;\">[latex]AB=\\begin{bmatrix} -7 & 10 \\\\ 30 & 11 \\end{bmatrix}\\not=\\begin{bmatrix} -9 & 10 & 10 \\\\ 4 & -8 & -12 \\\\ 10 & 4 & 21 \\end{bmatrix}=BA[\/latex]<\/p>\n<p>This illustrates the fact that matrix multiplication is not commutative.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Q: Is it possible for <em data-effect=\"italics\">AB<\/em> to be defined but not <em data-effect=\"italics\">BA<\/em>?<\/strong><\/p>\n<p><em>A: Yes, consider a matrix A with dimension and matrix B with dimension For the product AB the inner dimensions are 4 and the product is defined, but for the product BA the inner dimensions are 2 and 3 so the product is undefined.<\/em><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 10: Using Matrices in Real-World Problems<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Let\u2019s return to the problem presented at the opening of this section. We have Table 3, representing the equipment needs of two soccer teams.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 3<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 33.3333%;\"><\/td>\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Wildcats<\/strong><\/td>\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Mud Cats<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Goals<\/strong><\/td>\n<td style=\"width: 33.3333%; text-align: center;\">6<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">10<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Balls<\/strong><\/td>\n<td style=\"width: 33.3333%; text-align: center;\">30<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">24<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Jerseys<\/strong><\/td>\n<td style=\"width: 33.3333%; text-align: center;\">14<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">20<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We are also given the prices of the equipment, as shown in Table 4.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 4<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 50%; text-align: center;\"><strong>Goal<\/strong><\/td>\n<td style=\"width: 50%; text-align: center;\">$300<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\"><strong>Ball<\/strong><\/td>\n<td style=\"width: 50%; text-align: center;\">$10<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\"><strong>Jersey<\/strong><\/td>\n<td style=\"width: 50%; text-align: center;\">$30<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We will convert the data to matrices. Thus, the equipment need matrix is written as<\/p>\n<p style=\"text-align: center;\">[latex]E=\\begin{bmatrix} 6 & 10 \\\\ 30 & 24 \\\\ 14 & 20 \\end{bmatrix}[\/latex]<\/p>\n<p>The cost matrix is written as<\/p>\n<p>[latex]C=\\begin{bmatrix} 300 & 10 ^ 30 \\end{bmatrix}[\/latex]<\/p>\n<p>We perform matrix multiplication to obtain costs for the equipment.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll}\u00a0 C &=& \\begin{bmatrix} 300 & 10 & 30 \\end{bmatrix}\\begin{bmatrix} 6 & 10 \\\\ 30 & 24 \\\\ 14 & 20 \\end{bmatrix} \\\\ &=& \\begin{bmatrix} 300(6)+10(30)+30(14) & 300(10)+10(24)+30(20) \\end{bmatrix} \\\\ &=& \\begin{bmatrix} 2,520 & 3,840 \\end{bmatrix} \\end{array}[\/latex]<\/p>\n<p>The total cost for equipment for the Wildcats is $2,520, and the total cost for equipment for the Mud Cats is $3,840.<\/p>\n<section>\n<div><\/div>\n<\/section>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a matrix operation, evaluate using a calculator.<\/strong><\/p>\n<ol>\n<li>Save each matrix as a matrix variable [latex][A], [B], [C], \\ldots[\/latex]<\/li>\n<li>Enter the operation into the calculator, calling up each matrix variable as needed.<\/li>\n<li>If the operation is defined, the calculator will present the solution matrix; if the operation is undefined, it will display an error message.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 11: Using a Calculator to Perform Matrix Operations<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find [latex]AB-C[\/latex] given<\/p>\n<p style=\"text-align: center;\">[latex]A=\\begin{bmatrix} -15 & 25 & 32 \\\\ 41 & -7 & -28 \\\\ 10 & 34 & -2 \\end{bmatrix}, B=\\begin{bmatrix} 45 & 21 & -37 \\\\ -24 & 52 & 19 \\\\ 6 & -48 & -31 \\end{bmatrix}, \\quad \\text{and} \\quad C=\\begin{bmatrix} -100 & -89 & -98 \\\\ 25 & -56 & 74 \\\\ -67 & 42 & -75 \\end{bmatrix}.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>On the matrix page of the calculator, we enter matrix <em>A<\/em> above as the matrix variable [latex][A][\/latex] matrix <em>B<\/em> above as the matrix variable [latex][B][\/latex] and matrix <em>C<\/em> above as the matrix variable [latex][C]\/[\/latex]<\/p>\n<p>On the home screen of the calculator, we type in the problem and call up each matrix variable as needed.<\/p>\n<p style=\"text-align: center;\">[latex][A][B]-[C][\/latex]<\/p>\n<p>The calculator gives us the following matrix.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{bmatrix} -983 & -462 & 136 \\\\ 1,820 & 1,897 & -856 \\\\ -311 & 2,032 & 413 \\end{bmatrix}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Media<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1165137463912\">Access these online resources for additional instruction and practice with matrices and matrix operations.<\/p>\n<ul id=\"eip-id1165137681215\">\n<li><a href=\"http:\/\/openstax.org\/l\/matrixdimen\" target=\"_blank\" rel=\"noopener nofollow\">Dimensions of a Matrix<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/matrixaddsub\" target=\"_blank\" rel=\"noopener nofollow\">Matrix Addition and Subtraction<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/matrixoper\" target=\"_blank\" rel=\"noopener nofollow\">Matrix Operations<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/matrixmult\" target=\"_blank\" rel=\"noopener nofollow\">Matrix Multiplication<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">7.5 Section Exercises<\/span><\/h2>\n<section id=\"fs-id1165133245136\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1165134301522\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1165134301527\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134301529\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134301527-solution\">1<\/a><span class=\"os-divider\">. <\/span>Can we add any two matrices together? If so, explain why; if not, explain why not and give an example of two matrices that cannot be added together.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134183745\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134183746\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>Can we multiply any column matrix by any row matrix? Explain why or why not.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134183750\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134183752\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134183750-solution\">3<\/a><span class=\"os-divider\">. <\/span>Can both the products <em>AB<\/em> and <em>BA<\/em> be defined? If so, explain how; if not, explain why.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137629348\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137629349\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>Can any two matrices of the same size be multiplied? If so, explain why, and if not, explain why not and give an example of two matrices of the same size that cannot be multiplied together.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133088804\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133088805\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133088804-solution\">5<\/a><span class=\"os-divider\">. <\/span>Does matrix multiplication commute? That is, does [latex]AB=BA?[\/latex] If so, prove why it does. If not, explain why it does not.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165137770942\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id1165137770947\">For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\begin{bmatrix} 1 & 3 \\\\ 0 & 7 \\end{bmatrix}, B= \\begin{bmatrix} 2 & 14 \\\\ 22 & 6 \\end{bmatrix}, C= \\begin{bmatrix} 1 & 5 \\\\ 8 & 92 \\\\ 12 & 6 \\end{bmatrix}, D= \\begin{bmatrix} 10 & 14 \\\\ 7 & 2 \\\\ 5 & 61 \\end{bmatrix}, E= \\begin{bmatrix} 6 & 12 \\\\ 14 & 5 \\end{bmatrix}, F= \\begin{bmatrix} 0 & 9 \\\\ 78 & 17 \\\\ 15 & 4 \\end{bmatrix}[\/latex]<\/p>\n<div id=\"fs-id1165135381338\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134544655\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex]A+B[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165135481913\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135481914\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135481913-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex]C+D[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132035314\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132035315\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex]A+C[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135452967\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135452968\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135452967-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex]B-E[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135388909\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135388910\" data-type=\"problem\">\n<p><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex]C+F[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134589497\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134589498\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134589497-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex]D-B[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165133141433\">For the following exercises, use the matrices below to perform scalar multiplication.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\begin{bmatrix} 4 & 6 \\\\ 13 & 12 \\end{bmatrix}, B=\\begin{bmatrix} 3 & 9\\\\ 21 & 12 \\\\ 0 & 64 \\end{bmatrix}, C=\\begin{bmatrix} 16 & 3 & 7 & 18 \\\\ 90 & 5 & 3 & 29 \\end{bmatrix}, D=\\begin{bmatrix} 18 & 12 & 13 \\\\ 8 & 14 & 6 \\\\ 7 & 4 & 21 \\end{bmatrix}[\/latex]<\/p>\n<div id=\"fs-id1165135672778\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135672779\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex]5A[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134556459\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134556460\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134556459-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex]3B[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132935126\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132935127\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex]-2B[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135367787\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135367788\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135367787-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex]-4C[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135445702\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135445703\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex]\\frac{1}{2}C[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137896991\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137896992\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165137896991-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex]100D[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135256020\">For the following exercises, use the matrices below to perform matrix multiplication.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\begin{bmatrix} -1 & 5 \\\\ 3 & 2 \\end{bmatrix}, B=\\begin{bmatrix} 3 & 6 & 4 \\\\ -8 & 0 & 12 \\end{bmatrix}, C=\\begin{bmatrix} 4 & 10 \\\\ -2 & 6 \\\\ 5 & 9 \\end{bmatrix}, D=\\begin{bmatrix} 2 & -3 & 12 \\\\ 9 & 3 & 1 \\\\ 0 & 8 & -10 \\end{bmatrix}[\/latex]<\/p>\n<div id=\"fs-id1165132332419\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132332420\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex]AB[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133103836\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133103837\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133103836-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex]BC[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135649504\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135649505\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex]CA[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134282176\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134282177\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134282176-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex]BD[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133355960\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137736220\" data-type=\"problem\">\n<p><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex]DC[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135367683\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135367684\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135367683-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex]CB[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165134381432\">For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\begin{bmatrix} 2 & -5 \\\\ 6 &\u00a0 7 \\end{bmatrix}, B=\\begin{bmatrix}\u00a0 -9 & 6 \\\\ -4 & 2 \\end{bmatrix}, C=\\begin{bmatrix} 0 & 9 \\\\ 7 & 1 \\end{bmatrix}, D=\\begin{bmatrix} -8 & 7 & -5 \\\\ 4 & 3 & 2 \\\\ 0 & 9 & 2 \\end{bmatrix}, E=\\begin{bmatrix} 4 & 5 & 3 \\\\ 7 & -6 & -5 \\\\ 1 & 0 & 9 \\end{bmatrix}[\/latex]<\/p>\n<div id=\"fs-id1165134552652\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134552653\" data-type=\"problem\">\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex]A+B-C[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135320576\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135320577\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135320576-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex]4A+5D[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137683016\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137683017\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex]2C+B[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134042234\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134267898\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134042234-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex]3D+4E[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133176699\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133176700\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex]C-0.5D[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134042465\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134042466\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134042465-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex]100D-10E[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135196589\">For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: [latex]A^2=A\\cdot A[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]A=\\begin{bmatrix} -10 & 20 \\\\ 5 & 25 \\end{bmatrix}, B=\\begin{bmatrix}\u00a0 40 & 10 \\\\ -20 & 30 \\end{bmatrix}, C=\\begin{bmatrix} -1 & 0\\\\ 0 & -1 \\\\ 1 & 0 \\end{bmatrix}[\/latex]<\/p>\n<div id=\"fs-id1165135518214\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135518215\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">. [latex]AB[\/latex]<\/span><\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135332915\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135332916\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135332915-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex]BA[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165131958337\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165131958338\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex]CA[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137809962\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137809963\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165137809962-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex]BC[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135662475\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134112957\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex]A^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135440347\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135440348\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135440347-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex]B^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133157399\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133157400\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex]C^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133214671\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133214672\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133214671-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex]B^2A^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133324890\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133324891\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex]A^2B^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165131891720\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165131891721\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165131891720-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex](AB)^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135595090\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135595091\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex](BA)^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165131949634\">For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: [latex]A^2=A\\cdot A[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]A=\\begin{bmatrix} 1 & 0 \\\\ 2 & 3 \\end{bmatrix}, B=\\begin{bmatrix} -2 & 3 & 4 \\\\ -1 & 1 & -5 \\end{bmatrix}, C=\\begin{bmatrix} 0.5 & 0.1 \\\\ 1 & 0.2 \\\\ -0.5 & 0.3 \\end{bmatrix}, D=\\begin{bmatrix} 1 & 0 & -1 \\\\ -6 & 7 & 5 \\\\ 4 & 2 & 1 \\end{bmatrix}[\/latex]<\/p>\n<div id=\"fs-id1165135177605\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135177606\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135177605-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex]AB[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133092748\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133092749\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex]BA[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133134811\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133134812\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133134811-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex]BD[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135701628\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135701629\" data-type=\"problem\">\n<p><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex]DC[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135320241\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135320242\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135320241-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex]D^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135354776\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135354777\" data-type=\"problem\">\n<p><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex]A^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135421447\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135421448\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135421447-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex]D^3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135702617\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135702618\" data-type=\"problem\">\n<p><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex](AB)C[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135154488\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135154489\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135154488-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex]A(BC)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165137900018\" data-depth=\"2\">\n<h3 data-type=\"title\">Technology<\/h3>\n<p id=\"fs-id1165135455953\">For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\begin{bmatrix} -2 & 0 & 9 \\\\ 1 & 8 & -3 \\\\ 0.5 & 4 & 5 \\end{bmatrix}, B=\\begin{bmatrix} 0.5 & 3 & 0 \\\\ -4 & 1 & 6 \\\\ 8 & 7 & 2 \\end{bmatrix}, C=\\begin{bmatrix} 1 & 0 & 1 \\\\ 0 & 1 & 0\\\\ 1 & 0 & 1 \\end{bmatrix}[\/latex]<\/p>\n<div id=\"fs-id1165135369165\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135369166\" data-type=\"problem\">\n<p><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex]AB[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165135546873\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135546874\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135546873-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex]BA[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134407090\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134407091\" data-type=\"problem\">\n<p><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex]CA[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134038135\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134038136\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165134038135-solution\">53<\/a><span class=\"os-divider\">. <\/span> [latex]BC[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132986100\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132986101\" data-type=\"problem\">\n<p><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span> [latex]ABC[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165134188873\" data-depth=\"2\">\n<h3 data-type=\"title\">Extensions<\/h3>\n<p id=\"fs-id1165137407686\">For the following exercises, use the matrix below to perform the indicated operation on the given matrix.<\/p>\n<p style=\"text-align: center;\">[latex]B=\\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 1 & 0 \\end{bmatrix}[\/latex]<\/p>\n<div id=\"fs-id1165135298429\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135298430\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135298429-solution\">55<\/a><span class=\"os-divider\">. <\/span> [latex]B^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165133088779\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133088780\" data-type=\"problem\">\n<p><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span> [latex]B^3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133377041\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133377042\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165133377041-solution\">57<\/a><span class=\"os-divider\">. <\/span> [latex]B^4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134225694\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134225695\" data-type=\"problem\">\n<p><span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span> [latex]B^5[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135455009\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135455010\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-7\" data-page-slug=\"chapter-7\" data-page-uuid=\"f0a43272-7860-5760-ac27-5047923a0f08\" data-page-fragment=\"fs-id1165135455009-solution\">59<\/a><span class=\"os-divider\">. <\/span>Using the above questions, find a formula for [latex]B^n.[\/latex] Test the formula for [latex]B^{201}[\/latex] and [latex]B^{202},[\/latex] using a calculator.<\/p>\n<\/div>\n<\/section>\n<\/div>\n","protected":false},"author":158,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-216","chapter","type-chapter","status-publish","hentry"],"part":162,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/216","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/users\/158"}],"version-history":[{"count":21,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/216\/revisions"}],"predecessor-version":[{"id":1743,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/216\/revisions\/1743"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/parts\/162"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/216\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=216"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=216"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=216"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=216"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}