{"id":402,"date":"2019-07-15T22:44:46","date_gmt":"2019-07-15T22:44:46","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/chapter\/products-of-matrices\/"},"modified":"2019-07-15T22:44:46","modified_gmt":"2019-07-15T22:44:46","slug":"products-of-matrices","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/chapter\/products-of-matrices\/","title":{"raw":"Products of Matrices","rendered":"Products of Matrices"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Multiply a matrix by a scalar, sum scalar multiples of matrices.<\/li>\n \t<li>Multiply two matrices together.<\/li>\n \t<li>Use a calculator to perform operations on matrices.<\/li>\n<\/ul>\n<\/div>\nBesides adding and subtracting whole matrices, there are many situations in which we need to multiply a matrix by a constant called a scalar. A <strong>scalar<\/strong> is a real number quantity that has magnitude but not direction. For example, time, temperature, and distance are scalar quantities. The process of <strong>scalar multiplication<\/strong> 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.\n\nConsider 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 from the fall 2013 semester to the fall of 2014. They estimate that 15% more equipment is needed in both labs. The school\u2019s current inventory is displayed in the table below.\n<table style=\"height: 60px;\" summary=\"..\">\n<thead>\n<tr style=\"height: 15px;\">\n<th style=\"height: 15px;\"><\/th>\n<th style=\"height: 15px; text-align: center;\">Lab A<\/th>\n<th style=\"height: 15px; text-align: center;\">Lab B<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><strong>Computers<\/strong><\/td>\n<td style=\"height: 15px; text-align: center;\">15<\/td>\n<td style=\"height: 15px; text-align: center;\">27<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><strong>Computer Tables<\/strong><\/td>\n<td style=\"height: 15px; text-align: center;\">16<\/td>\n<td style=\"height: 15px; text-align: center;\">34<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><strong>Chairs<\/strong><\/td>\n<td style=\"height: 15px; text-align: center;\">16<\/td>\n<td style=\"height: 15px; text-align: center;\">34<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nConverting the data to a matrix, we have the computer inventory in fall 2013 given by\n<p style=\"text-align: center;\">[latex]{C}_{2013}=\\left[\\begin{array}{c}15\\\\ 16\\\\ 16\\end{array}\\begin{array}{c}27\\\\ 34\\\\ 34\\end{array}\\right][\/latex]<\/p>\nTo calculate how much computer equipment will be needed in 2014, we multiply all entries in matrix [latex]C[\/latex] by 0.15.\n<p style=\"text-align: center;\">[latex]\\left(0.15\\right){C}_{2013}=\\left[\\begin{array}{c}\\left(0.15\\right)15\\\\ \\left(0.15\\right)16\\\\ \\left(0.15\\right)16\\end{array}\\begin{array}{c}\\left(0.15\\right)27\\\\ \\left(0.15\\right)34\\\\ \\left(0.15\\right)34\\end{array}\\right]=\\left[\\begin{array}{c}2.25\\\\ 2.4\\\\ 2.4\\end{array}\\begin{array}{c}4.05\\\\ 5.1\\\\ 5.1\\end{array}\\right][\/latex]<\/p>\nWe must round up to the next integer, so the amount of new equipment needed is\n<p style=\"text-align: center;\">[latex]\\left[\\begin{array}{c}3\\\\ 3\\\\ 3\\end{array}\\begin{array}{c}5\\\\ 6\\\\ 6\\end{array}\\right][\/latex]<\/p>\nAdding the two matrices as shown below, we see the new inventory amounts.\n<p style=\"text-align: center;\">[latex]\\left[\\begin{array}{c}15\\\\ 16\\\\ 16\\end{array}\\begin{array}{c}27\\\\ 34\\\\ 34\\end{array}\\right]+\\left[\\begin{array}{c}3\\\\ 3\\\\ 3\\end{array}\\begin{array}{c}5\\\\ 6\\\\ 6\\end{array}\\right]=\\left[\\begin{array}{c}18\\\\ 19\\\\ 19\\end{array}\\begin{array}{c}32\\\\ 40\\\\ 40\\end{array}\\right][\/latex]<\/p>\nThis means\n<p style=\"text-align: center;\">[latex]{C}_{2014}=\\left[\\begin{array}{c}18\\\\ 19\\\\ 19\\end{array}\\begin{array}{c}32\\\\ 40\\\\ 40\\end{array}\\right][\/latex]<\/p>\nThus, Lab A will have 18 computers, 19 computer tables, and 19 chairs; Lab B will have 32 computers, 40 computer tables, and 40 chairs.\n<div class=\"textbox\">\n<h3>A General Note: Scalar Multiplication<\/h3>\nScalar multiplication involves finding the product of a constant by each entry in the matrix. Given\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cccc}{a}_{11}&amp; &amp; &amp; {a}_{12}\\\\ {a}_{21}&amp; &amp; &amp; {a}_{22}\\end{array}\\right][\/latex]<\/p>\nthe scalar multiple [latex]cA[\/latex] is\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}cA &amp; =c\\left[\\begin{array}{ccc}{a}_{11}&amp; &amp; {a}_{12}\\\\ {a}_{21}&amp; &amp; {a}_{22}\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{ccc}c{a}_{11}&amp; &amp; c{a}_{12}\\\\ c{a}_{21}&amp; &amp; c{a}_{22}\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\nScalar multiplication is distributive. For the matrices [latex]A,B[\/latex], and [latex]C[\/latex] with scalars [latex]a[\/latex] and [latex]b[\/latex],\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\\\ \\begin{array}{c}a\\left(A+B\\right)=aA+aB\\\\ \\left(a+b\\right)A=aA+bA\\end{array}\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Multiplying the Matrix by a Scalar<\/h3>\nMultiply matrix [latex]A[\/latex] by the scalar 3.\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}8&amp; 1\\\\ 5&amp; 4\\end{array}\\right][\/latex]<\/p>\n[reveal-answer q=\"90744\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"90744\"]\n\nMultiply each entry in [latex]A[\/latex] by the scalar 3.\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}3A &amp; =3\\left[\\begin{array}{rr}\\hfill 8&amp; \\hfill 1\\\\ \\hfill 5&amp; \\hfill 4\\end{array}\\right]\\hfill \\\\ &amp; = \\left[\\begin{array}{rr}\\hfill 3\\cdot 8&amp; \\hfill 3\\cdot 1\\\\ \\hfill 3\\cdot 5&amp; \\hfill 3\\cdot 4\\end{array}\\right]\\hfill \\\\ &amp; = \\left[\\begin{array}{rr}\\hfill 24&amp; \\hfill 3\\\\ \\hfill 15&amp; \\hfill 12\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nGiven matrix [latex]B,\\text{}[\/latex] find [latex]-2B[\/latex] where\n<p style=\"text-align: center;\">[latex]B=\\left[\\begin{array}{cc}4&amp; 1\\\\ 3&amp; 2\\end{array}\\right][\/latex]<\/p>\n[reveal-answer q=\"2999\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"2999\"][latex]-2B=\\left[\\begin{array}{cc}-8&amp; -2\\\\ -6&amp; -4\\end{array}\\right][\/latex][\/hidden-answer]\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Sum of Scalar Multiples<\/h3>\nFind the sum [latex]3A+2B[\/latex].\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{rrr}\\hfill 1&amp; \\hfill -2&amp; \\hfill 0\\\\ \\hfill 0&amp; \\hfill -1&amp; \\hfill 2\\\\ \\hfill 4&amp; \\hfill 3&amp; \\hfill -6\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{rrr}\\hfill -1&amp; \\hfill 2&amp; \\hfill 1\\\\ \\hfill 0&amp; \\hfill -3&amp; \\hfill 2\\\\ \\hfill 0&amp; \\hfill 1&amp; \\hfill -4\\end{array}\\right][\/latex]<\/p>\n[reveal-answer q=\"755068\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"755068\"]\n\nFirst, find [latex]3A,\\text{}[\/latex] then [latex]2B[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ 3A &amp; =\\left[\\begin{array}{lll}3\\cdot 1\\hfill &amp; 3\\left(-2\\right)\\hfill &amp; 3\\cdot 0\\hfill \\\\ 3\\cdot 0\\hfill &amp; 3\\left(-1\\right)\\hfill &amp; 3\\cdot 2\\hfill \\\\ 3\\cdot 4\\hfill &amp; 3\\cdot 3\\hfill &amp; 3\\left(-6\\right)\\hfill \\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{rrr}\\hfill 3&amp; \\hfill -6&amp; \\hfill 0\\\\ \\hfill 0&amp; \\hfill -3&amp; \\hfill 6\\\\ \\hfill 12&amp; \\hfill 9&amp; \\hfill -18\\end{array}\\right]\\hfill \\end{array}[\/latex]\n[latex]\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ 2B &amp; =\\left[\\begin{array}{lll}2\\left(-1\\right)\\hfill &amp; 2\\cdot 2\\hfill &amp; 2\\cdot 1\\hfill \\\\ 2\\cdot 0\\hfill &amp; 2\\left(-3\\right)\\hfill &amp; 2\\cdot 2\\hfill \\\\ 2\\cdot 0\\hfill &amp; 2\\cdot 1\\hfill &amp; 2\\left(-4\\right)\\hfill \\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{rrr}\\hfill -2&amp; \\hfill 4&amp; \\hfill 2\\\\ \\hfill 0&amp; \\hfill -6&amp; \\hfill 4\\\\ \\hfill 0&amp; \\hfill 2&amp; \\hfill -8\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\nNow, add [latex]3A+2B[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ 3A+2B &amp; =\\left[\\begin{array}{rrr}\\hfill 3&amp; \\hfill -6&amp; \\hfill 0\\\\ \\hfill 0&amp; \\hfill -3&amp; \\hfill 6\\\\ \\hfill 12&amp; \\hfill 9&amp; \\hfill -18\\end{array}\\right]+\\left[\\begin{array}{rrr}\\hfill -2&amp; \\hfill 4&amp; \\hfill 2\\\\ \\hfill 0&amp; \\hfill -6&amp; \\hfill 4\\\\ \\hfill 0&amp; \\hfill 2&amp; \\hfill -8\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{rrr}\\hfill 3 - 2&amp; \\hfill -6+4&amp; \\hfill 0+2\\\\ \\hfill 0+0&amp; \\hfill -3 - 6&amp; \\hfill 6+4\\\\ \\hfill 12+0&amp; \\hfill 9+2&amp; \\hfill -18 - 8\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{rrr}\\hfill 1&amp; \\hfill -2&amp; \\hfill 2\\\\ \\hfill 0&amp; \\hfill -9&amp; \\hfill 10\\\\ \\hfill 12&amp; \\hfill 11&amp; \\hfill -26\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n[ohm_question]6386[\/ohm_question]\n\n<\/div>\n&nbsp;\n<h2 style=\"text-align: left;\">Finding the Product of Two Matrices<\/h2>\nIn addition to multiplying a matrix by a scalar, we can multiply two matrices. Finding the <strong>product of two matrices<\/strong> is only possible when the <em>inner dimensions<\/em> 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 [latex]A[\/latex] is an [latex]\\text{ }m\\text{ }\\times \\text{ }r\\text{ }[\/latex] matrix and [latex]B[\/latex] is an [latex]\\text{ }r\\text{ }\\times \\text{ }n\\text{ }[\/latex] matrix, then the product matrix [latex]AB[\/latex] is an [latex]\\text{ }m\\text{ }\\times \\text{ }n\\text{ }[\/latex] matrix. For example, the product [latex]AB[\/latex] is possible because the number of columns in [latex]A[\/latex] is the same as the number of rows in [latex]B[\/latex]. If the inner dimensions do not match, the product is not defined.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03193518\/12.jpg\" alt=\"A has two rows and three columns and B has three rows and three columns. Because the number of columns in A matches the number of rows in B, the product of A and B is defined.\" width=\"154\" height=\"93\">\n\nWe multiply entries of [latex]A[\/latex] with entries of [latex]B[\/latex] according to a specific pattern as outlined below. The process of <strong>matrix multiplication<\/strong> becomes clearer when working a problem with real numbers.\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\nWork through the example model of matrix multiplication below on paper, then apply the process to the example problem below. It may take more than once to gain familiarity with it. Don't be discouraged if you don't understand fully right away. Matrix multiplication is a new skill and it will take time and practice for it to feel comfortable.\n\n<\/div>\nTo obtain the entries in row [latex]i[\/latex] of [latex]AB,\\text{}[\/latex] we multiply the entries in row [latex]i[\/latex] of [latex]A[\/latex] by column [latex]j[\/latex] in [latex]B[\/latex] and add. For example, given matrices [latex]A[\/latex] and [latex]B,\\text{}[\/latex] where the dimensions of [latex]A[\/latex] are [latex]2\\text{ }\\times \\text{ }3[\/latex] and the dimensions of [latex]B[\/latex] are [latex]3\\text{ }\\times \\text{ }3,\\text{}[\/latex] the product of [latex]AB[\/latex] will be a [latex]2\\text{ }\\times \\text{ }3[\/latex] matrix.\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{rrr}\\hfill {a}_{11}&amp; \\hfill {a}_{12}&amp; \\hfill {a}_{13}\\\\ \\hfill {a}_{21}&amp; \\hfill {a}_{22}&amp; \\hfill {a}_{23}\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{rrr}\\hfill {b}_{11}&amp; \\hfill {b}_{12}&amp; \\hfill {b}_{13}\\\\ \\hfill {b}_{21}&amp; \\hfill {b}_{22}&amp; \\hfill {b}_{23}\\\\ \\hfill {b}_{31}&amp; \\hfill {b}_{32}&amp; \\hfill {b}_{33}\\end{array}\\right][\/latex]<\/p>\nMultiply and add as follows to obtain the first entry of the product matrix [latex]AB[\/latex].\n<ol>\n \t<li>To obtain the entry in row 1, column 1 of [latex]AB,\\text{}[\/latex] multiply the first row in [latex]A[\/latex] by the first column in [latex]B[\/latex] and add.\n<div style=\"text-align: center;\">[latex]\\left[\\begin{array}{ccc}{a}_{11}&amp; {a}_{12}&amp; {a}_{13}\\end{array}\\right]\\cdot \\left[\\begin{array}{c}{b}_{11}\\\\ {b}_{21}\\\\ {b}_{31}\\end{array}\\right]={a}_{11}\\cdot {b}_{11}+{a}_{12}\\cdot {b}_{21}+{a}_{13}\\cdot {b}_{31}[\/latex]<\/div><\/li>\n \t<li>To obtain the entry in row 1, column 2 of [latex]AB,\\text{}[\/latex] multiply the first row of [latex]A[\/latex] by the second column in [latex]B[\/latex] and add.\n<div style=\"text-align: center;\">[latex]\\left[\\begin{array}{ccc}{a}_{11}&amp; {a}_{12}&amp; {a}_{13}\\end{array}\\right]\\cdot \\left[\\begin{array}{c}{b}_{12}\\\\ {b}_{22}\\\\ {b}_{32}\\end{array}\\right]={a}_{11}\\cdot {b}_{12}+{a}_{12}\\cdot {b}_{22}+{a}_{13}\\cdot {b}_{32}[\/latex]<\/div><\/li>\n \t<li>To obtain the entry in row 1, column 3 of [latex]AB,\\text{}[\/latex] multiply the first row of [latex]A[\/latex] by the third column in [latex]B[\/latex] and add.\n<div style=\"text-align: center;\">[latex]\\left[\\begin{array}{ccc}{a}_{11}&amp; {a}_{12}&amp; {a}_{13}\\end{array}\\right]\\cdot \\left[\\begin{array}{c}{b}_{13}\\\\ {b}_{23}\\\\ {b}_{33}\\end{array}\\right]={a}_{11}\\cdot {b}_{13}+{a}_{12}\\cdot {b}_{23}+{a}_{13}\\cdot {b}_{33}[\/latex]<\/div><\/li>\n<\/ol>\nWe proceed the same way to obtain the second row of [latex]AB[\/latex]. In other words, row 2 of [latex]A[\/latex] times column 1 of [latex]B[\/latex]; row 2 of [latex]A[\/latex] times column 2 of [latex]B[\/latex]; row 2 of [latex]A[\/latex] times column 3 of [latex]B[\/latex]. When complete, the product matrix will be\n<p style=\"text-align: center;\">[latex]AB=\\left[\\begin{array}{c}\\begin{array}{l}{a}_{11}\\cdot {b}_{11}+{a}_{12}\\cdot {b}_{21}+{a}_{13}\\cdot {b}_{31}\\\\ \\end{array}\\\\ {a}_{21}\\cdot {b}_{11}+{a}_{22}\\cdot {b}_{21}+{a}_{23}\\cdot {b}_{31}\\end{array}\\begin{array}{c}\\begin{array}{l}{a}_{11}\\cdot {b}_{12}+{a}_{12}\\cdot {b}_{22}+{a}_{13}\\cdot {b}_{32}\\\\ \\end{array}\\\\ {a}_{21}\\cdot {b}_{12}+{a}_{22}\\cdot {b}_{22}+{a}_{23}\\cdot {b}_{32}\\end{array}\\begin{array}{c}\\begin{array}{l}{a}_{11}\\cdot {b}_{13}+{a}_{12}\\cdot {b}_{23}+{a}_{13}\\cdot {b}_{33}\\\\ \\end{array}\\\\ {a}_{21}\\cdot {b}_{13}+{a}_{22}\\cdot {b}_{23}+{a}_{23}\\cdot {b}_{33}\\end{array}\\right][\/latex]<\/p>\n\n<div class=\"textbox\">\n<h3>A General Note: Properties of Matrix Multiplication<\/h3>\nFor the matrices [latex]A,B,\\text{}[\/latex] and [latex]C[\/latex] the following properties hold.\n<ul>\n \t<li>Matrix multiplication is associative:\n<div style=\"text-align: center;\">[latex]\\left(AB\\right)C=A\\left(BC\\right)[\/latex]<\/div><\/li>\n \t<li>Matrix multiplication is distributive:\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ C\\left(A+B\\right)=CA+CB,\\end{array}\\hfill \\\\ \\left(A+B\\right)C=AC+BC.\\hfill \\end{array}[\/latex]<\/div><\/li>\n<\/ul>\nNote that matrix multiplication is not commutative.\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Multiplying Two Matrices<\/h3>\nMultiply matrix [latex]A[\/latex] and matrix [latex]B[\/latex].\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}1&amp; 2\\\\ 3&amp; 4\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{cc}5&amp; 6\\\\ 7&amp; 8\\end{array}\\right][\/latex]<\/p>\n[reveal-answer q=\"843176\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"843176\"]\n\nFirst, we check the dimensions of the matrices. Matrix [latex]A[\/latex] has dimensions [latex]2\\times 2[\/latex] and matrix [latex]B[\/latex] 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].\n\nWe perform the operations outlined previously.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03193520\/CNX_Precalc_Figure_09_05_0032.jpg\" alt=\"The first column of the product of A and B is defined as the result of matrix -vector multiplication of A and the first column of B. Column two of the product of A and B is defined as the result of the matrix-vector multiplication of A and the second column of B.\" width=\"487\" height=\"211\">\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question]1075[\/ohm_question]\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Multiplying Two Matrices<\/h3>\nGiven [latex]A[\/latex] and [latex]B:[\/latex]\n<ol>\n \t<li>Find [latex]AB[\/latex].<\/li>\n \t<li>Find [latex]BA[\/latex].<\/li>\n<\/ol>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}\\hfill -1&amp; \\hfill 2&amp; \\hfill 3\\\\ \\hfill 4&amp; \\hfill 0&amp; \\hfill 5\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{cc}\\hfill 5&amp; \\hfill -1\\\\ \\hfill -4&amp; \\hfill 0\\\\ \\hfill 2&amp; \\hfill 3\\end{array}\\right][\/latex]<\/p>\n[reveal-answer q=\"403957\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"403957\"]\n<ol>\n \t<li>As the dimensions of [latex]A[\/latex] are [latex]2\\text{}\\times \\text{}3[\/latex] and the dimensions of [latex]B[\/latex] are [latex]3\\text{}\\times \\text{}2,\\text{}[\/latex] these matrices can be multiplied together because the number of columns in [latex]A[\/latex] matches the number of rows in [latex]B[\/latex]. The resulting product will be a [latex]2\\text{}\\times \\text{}2[\/latex] matrix, the number of rows in [latex]A[\/latex] by the number of columns in [latex]B[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\hfill \\\\ AB &amp; =\\left[\\begin{array}{rrr}\\hfill -1&amp; \\hfill 2&amp; \\hfill 3\\\\ \\hfill 4&amp; \\hfill 0&amp; \\hfill 5\\end{array}\\right]\\text{ }\\left[\\begin{array}{rr}\\hfill 5&amp; \\hfill -1\\\\ \\hfill -4&amp; \\hfill 0\\\\ \\hfill 2&amp; \\hfill 3\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{rr}\\hfill -1\\left(5\\right)+2\\left(-4\\right)+3\\left(2\\right)&amp; \\hfill -1\\left(-1\\right)+2\\left(0\\right)+3\\left(3\\right)\\\\ \\hfill 4\\left(5\\right)+0\\left(-4\\right)+5\\left(2\\right)&amp; \\hfill 4\\left(-1\\right)+0\\left(0\\right)+5\\left(3\\right)\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{rr}\\hfill -7&amp; \\hfill 10\\\\ \\hfill 30&amp; \\hfill 11\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/div><\/li>\n \t<li>The dimensions of [latex]B[\/latex] are [latex]3\\times 2[\/latex] and the dimensions of [latex]A[\/latex] 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.\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\hfill \\\\ BA &amp; =\\left[\\begin{array}{rr}\\hfill 5&amp; \\hfill -1\\\\ \\hfill -4&amp; \\hfill 0\\\\ \\hfill 2&amp; \\hfill 3\\end{array}\\right]\\text{ }\\left[\\begin{array}{rrr}\\hfill -1&amp; \\hfill 2&amp; \\hfill 3\\\\ \\hfill 4&amp; \\hfill 0&amp; \\hfill 5\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{rrr}\\hfill 5\\left(-1\\right)+-1\\left(4\\right)&amp; \\hfill 5\\left(2\\right)+-1\\left(0\\right)&amp; \\hfill 5\\left(3\\right)+-1\\left(5\\right)\\\\ \\hfill -4\\left(-1\\right)+0\\left(4\\right)&amp; \\hfill -4\\left(2\\right)+0\\left(0\\right)&amp; \\hfill -4\\left(3\\right)+0\\left(5\\right)\\\\ \\hfill 2\\left(-1\\right)+3\\left(4\\right)&amp; \\hfill 2\\left(2\\right)+3\\left(0\\right)&amp; \\hfill 2\\left(3\\right)+3\\left(5\\right)\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{rrr}\\hfill -9&amp; \\hfill 10&amp; \\hfill 10\\\\ \\hfill 4&amp; \\hfill -8&amp; \\hfill -12\\\\ \\hfill 10&amp; \\hfill 4&amp; \\hfill 21\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/div><\/li>\n<\/ol>\n<h4>Analysis of the Solution<\/h4>\nNotice that the products [latex]AB[\/latex] and [latex]BA[\/latex] are not equal.\n<p style=\"text-align: center;\">[latex]AB=\\left[\\begin{array}{cc}-7&amp; 10\\\\ 30&amp; 11\\end{array}\\right]\\ne \\left[\\begin{array}{ccc}-9&amp; 10&amp; 10\\\\ 4&amp; -8&amp; -12\\\\ 10&amp; 4&amp; 21\\end{array}\\right]=BA[\/latex]<\/p>\nThis illustrates the fact that matrix multiplication is not commutative.\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<strong>Is it possible for AB to be defined but not BA?<\/strong>\n\n<em>Yes, consider a matrix A with dimension [latex]3\\times 4[\/latex] and matrix B with dimension [latex]4\\times 2[\/latex]. 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>\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using Matrices in Real-World Problems<\/h3>\nLet\u2019s return to the problem presented at the opening of this section. We have the table below, representing the equipment needs of two soccer teams.\n<table summary=\"..\">\n<thead>\n<tr>\n<th><\/th>\n<th style=\"text-align: center;\">Wildcats<\/th>\n<th style=\"text-align: center;\">Mud Cats<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><strong>Goals<\/strong><\/td>\n<td style=\"text-align: center;\">6<\/td>\n<td style=\"text-align: center;\">10<\/td>\n<\/tr>\n<tr>\n<td><strong>Balls<\/strong><\/td>\n<td style=\"text-align: center;\">30<\/td>\n<td style=\"text-align: center;\">24<\/td>\n<\/tr>\n<tr>\n<td><strong>Jerseys<\/strong><\/td>\n<td style=\"text-align: center;\">14<\/td>\n<td style=\"text-align: center;\">20<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nWe are also given the prices of the equipment, as shown in the table below.\n<table summary=\"..\">\n<tbody>\n<tr>\n<td><strong>Goal<\/strong><\/td>\n<td>$300<\/td>\n<\/tr>\n<tr>\n<td><strong>Ball<\/strong><\/td>\n<td>$10<\/td>\n<\/tr>\n<tr>\n<td><strong>Jersey<\/strong><\/td>\n<td>$30<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nWe will convert the data to matrices. Thus, the equipment need matrix is written as\n<p style=\"text-align: center;\">[latex]E=\\left[\\begin{array}{c}6\\\\ 30\\\\ 14\\end{array}\\begin{array}{c}10\\\\ 24\\\\ 20\\end{array}\\right][\/latex]<\/p>\nThe cost matrix is written as\n<p style=\"text-align: center;\">[latex]C=\\left[\\begin{array}{ccc}300&amp; 10&amp; 30\\end{array}\\right][\/latex]\nWe perform matrix multiplication to obtain costs for the equipment.\n[latex]\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ CE &amp; =\\left[\\begin{array}{rrr}\\hfill 300&amp; \\hfill 10&amp; \\hfill 30\\end{array}\\right]\\cdot \\left[\\begin{array}{rr}\\hfill 6&amp; \\hfill 10\\\\ \\hfill 30&amp; \\hfill 24\\\\ \\hfill 14&amp; \\hfill 20\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{rr}\\hfill 300\\left(6\\right)+10\\left(30\\right)+30\\left(14\\right)&amp; \\hfill 300\\left(10\\right)+10\\left(24\\right)+30\\left(20\\right)\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{rr}\\hfill 2,520&amp; \\hfill 3,840\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\nThe total cost for equipment for the Wildcats is $2,520, and the total cost for equipment for the Mud Cats is $3,840.\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a matrix operation, evaluate using a calculator<strong>\n<\/strong><\/h3>\n<ol>\n \t<li>Save each matrix as a matrix variable\n<div style=\"text-align: center;\">[latex]\\left[A\\right],\\left[B\\right],\\left[C\\right],..[\/latex]<\/div><\/li>\n \t<li>Enter the operation into the calculator, calling up each matrix variable as needed.<\/li>\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>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using a Calculator to Perform Matrix Operations<\/h3>\nFind [latex]AB-C[\/latex]&nbsp;given\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{rrr}\\hfill -15&amp; \\hfill 25&amp; \\hfill 32\\\\ \\hfill 41&amp; \\hfill -7&amp; \\hfill -28\\\\ \\hfill 10&amp; \\hfill 34&amp; \\hfill -2\\end{array}\\right],B=\\left[\\begin{array}{rrr}\\hfill 45&amp; \\hfill 21&amp; \\hfill -37\\\\ \\hfill -24&amp; \\hfill 52&amp; \\hfill 19\\\\ \\hfill 6&amp; \\hfill -48&amp; \\hfill -31\\end{array}\\right],\\text{and }C=\\left[\\begin{array}{rrr}\\hfill -100&amp; \\hfill -89&amp; \\hfill -98\\\\ \\hfill 25&amp; \\hfill -56&amp; \\hfill 74\\\\ \\hfill -67&amp; \\hfill 42&amp; \\hfill -75\\end{array}\\right][\/latex].<\/p>\n[reveal-answer q=\"32907\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"32907\"]\n\nOn the matrix page of the calculator, we enter matrix [latex]A[\/latex] above as the matrix variable [latex]\\left[A\\right][\/latex], matrix [latex]B[\/latex] above as the matrix variable [latex]\\left[B\\right][\/latex], and matrix [latex]C[\/latex] above as the matrix variable [latex]\\left[C\\right][\/latex].\n\nOn the home screen of the calculator, we type in the problem and call up each matrix variable as needed.\n<p style=\"text-align: center;\">[latex]\\left[A\\right]\\times \\left[B\\right]-\\left[C\\right][\/latex]\nThe calculator gives us the following matrix.\n[latex]\\left[\\begin{array}{rrr}\\hfill -983&amp; \\hfill -462&amp; \\hfill 136\\\\ \\hfill 1,820&amp; \\hfill 1,897&amp; \\hfill -856\\\\ \\hfill -311&amp; \\hfill 2,032&amp; \\hfill 413\\end{array}\\right][\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Multiply a matrix by a scalar, sum scalar multiples of matrices.<\/li>\n<li>Multiply two matrices together.<\/li>\n<li>Use a calculator to perform operations on matrices.<\/li>\n<\/ul>\n<\/div>\n<p>Besides adding and subtracting whole matrices, there are many situations in which we need to multiply a matrix by a constant called a scalar. A <strong>scalar<\/strong> is a real number quantity that has magnitude but not direction. For example, time, temperature, and distance are scalar quantities. The process of <strong>scalar multiplication<\/strong> 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>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 from the fall 2013 semester to the fall of 2014. They estimate that 15% more equipment is needed in both labs. The school\u2019s current inventory is displayed in the table below.<\/p>\n<table style=\"height: 60px;\" summary=\"..\">\n<thead>\n<tr style=\"height: 15px;\">\n<th style=\"height: 15px;\"><\/th>\n<th style=\"height: 15px; text-align: center;\">Lab A<\/th>\n<th style=\"height: 15px; text-align: center;\">Lab B<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><strong>Computers<\/strong><\/td>\n<td style=\"height: 15px; text-align: center;\">15<\/td>\n<td style=\"height: 15px; text-align: center;\">27<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><strong>Computer Tables<\/strong><\/td>\n<td style=\"height: 15px; text-align: center;\">16<\/td>\n<td style=\"height: 15px; text-align: center;\">34<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><strong>Chairs<\/strong><\/td>\n<td style=\"height: 15px; text-align: center;\">16<\/td>\n<td style=\"height: 15px; text-align: center;\">34<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Converting the data to a matrix, we have the computer inventory in fall 2013 given by<\/p>\n<p style=\"text-align: center;\">[latex]{C}_{2013}=\\left[\\begin{array}{c}15\\\\ 16\\\\ 16\\end{array}\\begin{array}{c}27\\\\ 34\\\\ 34\\end{array}\\right][\/latex]<\/p>\n<p>To calculate how much computer equipment will be needed in 2014, we multiply all entries in matrix [latex]C[\/latex] by 0.15.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(0.15\\right){C}_{2013}=\\left[\\begin{array}{c}\\left(0.15\\right)15\\\\ \\left(0.15\\right)16\\\\ \\left(0.15\\right)16\\end{array}\\begin{array}{c}\\left(0.15\\right)27\\\\ \\left(0.15\\right)34\\\\ \\left(0.15\\right)34\\end{array}\\right]=\\left[\\begin{array}{c}2.25\\\\ 2.4\\\\ 2.4\\end{array}\\begin{array}{c}4.05\\\\ 5.1\\\\ 5.1\\end{array}\\right][\/latex]<\/p>\n<p>We must round up to the next integer, so the amount of new equipment needed is<\/p>\n<p style=\"text-align: center;\">[latex]\\left[\\begin{array}{c}3\\\\ 3\\\\ 3\\end{array}\\begin{array}{c}5\\\\ 6\\\\ 6\\end{array}\\right][\/latex]<\/p>\n<p>Adding the two matrices as shown below, we see the new inventory amounts.<\/p>\n<p style=\"text-align: center;\">[latex]\\left[\\begin{array}{c}15\\\\ 16\\\\ 16\\end{array}\\begin{array}{c}27\\\\ 34\\\\ 34\\end{array}\\right]+\\left[\\begin{array}{c}3\\\\ 3\\\\ 3\\end{array}\\begin{array}{c}5\\\\ 6\\\\ 6\\end{array}\\right]=\\left[\\begin{array}{c}18\\\\ 19\\\\ 19\\end{array}\\begin{array}{c}32\\\\ 40\\\\ 40\\end{array}\\right][\/latex]<\/p>\n<p>This means<\/p>\n<p style=\"text-align: center;\">[latex]{C}_{2014}=\\left[\\begin{array}{c}18\\\\ 19\\\\ 19\\end{array}\\begin{array}{c}32\\\\ 40\\\\ 40\\end{array}\\right][\/latex]<\/p>\n<p>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\">\n<h3>A General Note: Scalar Multiplication<\/h3>\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=\\left[\\begin{array}{cccc}{a}_{11}& & & {a}_{12}\\\\ {a}_{21}& & & {a}_{22}\\end{array}\\right][\/latex]<\/p>\n<p>the scalar multiple [latex]cA[\/latex] is<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}cA & =c\\left[\\begin{array}{ccc}{a}_{11}& & {a}_{12}\\\\ {a}_{21}& & {a}_{22}\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{ccc}c{a}_{11}& & c{a}_{12}\\\\ c{a}_{21}& & c{a}_{22}\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\n<p>Scalar multiplication is distributive. For the matrices [latex]A,B[\/latex], and [latex]C[\/latex] with scalars [latex]a[\/latex] and [latex]b[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\\\ \\begin{array}{c}a\\left(A+B\\right)=aA+aB\\\\ \\left(a+b\\right)A=aA+bA\\end{array}\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Multiplying the Matrix by a Scalar<\/h3>\n<p>Multiply matrix [latex]A[\/latex] by the scalar 3.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}8& 1\\\\ 5& 4\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q90744\">Show Solution<\/span><\/p>\n<div id=\"q90744\" class=\"hidden-answer\" style=\"display: none\">\n<p>Multiply each entry in [latex]A[\/latex] by the scalar 3.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}3A & =3\\left[\\begin{array}{rr}\\hfill 8& \\hfill 1\\\\ \\hfill 5& \\hfill 4\\end{array}\\right]\\hfill \\\\ & = \\left[\\begin{array}{rr}\\hfill 3\\cdot 8& \\hfill 3\\cdot 1\\\\ \\hfill 3\\cdot 5& \\hfill 3\\cdot 4\\end{array}\\right]\\hfill \\\\ & = \\left[\\begin{array}{rr}\\hfill 24& \\hfill 3\\\\ \\hfill 15& \\hfill 12\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given matrix [latex]B,\\text{}[\/latex] find [latex]-2B[\/latex] where<\/p>\n<p style=\"text-align: center;\">[latex]B=\\left[\\begin{array}{cc}4& 1\\\\ 3& 2\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q2999\">Show Solution<\/span><\/p>\n<div id=\"q2999\" class=\"hidden-answer\" style=\"display: none\">[latex]-2B=\\left[\\begin{array}{cc}-8& -2\\\\ -6& -4\\end{array}\\right][\/latex]<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Sum of Scalar Multiples<\/h3>\n<p>Find the sum [latex]3A+2B[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{rrr}\\hfill 1& \\hfill -2& \\hfill 0\\\\ \\hfill 0& \\hfill -1& \\hfill 2\\\\ \\hfill 4& \\hfill 3& \\hfill -6\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{rrr}\\hfill -1& \\hfill 2& \\hfill 1\\\\ \\hfill 0& \\hfill -3& \\hfill 2\\\\ \\hfill 0& \\hfill 1& \\hfill -4\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q755068\">Show Solution<\/span><\/p>\n<div id=\"q755068\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, find [latex]3A,\\text{}[\/latex] then [latex]2B[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ 3A & =\\left[\\begin{array}{lll}3\\cdot 1\\hfill & 3\\left(-2\\right)\\hfill & 3\\cdot 0\\hfill \\\\ 3\\cdot 0\\hfill & 3\\left(-1\\right)\\hfill & 3\\cdot 2\\hfill \\\\ 3\\cdot 4\\hfill & 3\\cdot 3\\hfill & 3\\left(-6\\right)\\hfill \\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rrr}\\hfill 3& \\hfill -6& \\hfill 0\\\\ \\hfill 0& \\hfill -3& \\hfill 6\\\\ \\hfill 12& \\hfill 9& \\hfill -18\\end{array}\\right]\\hfill \\end{array}[\/latex]<br \/>\n[latex]\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ 2B & =\\left[\\begin{array}{lll}2\\left(-1\\right)\\hfill & 2\\cdot 2\\hfill & 2\\cdot 1\\hfill \\\\ 2\\cdot 0\\hfill & 2\\left(-3\\right)\\hfill & 2\\cdot 2\\hfill \\\\ 2\\cdot 0\\hfill & 2\\cdot 1\\hfill & 2\\left(-4\\right)\\hfill \\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rrr}\\hfill -2& \\hfill 4& \\hfill 2\\\\ \\hfill 0& \\hfill -6& \\hfill 4\\\\ \\hfill 0& \\hfill 2& \\hfill -8\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\n<p>Now, add [latex]3A+2B[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ 3A+2B & =\\left[\\begin{array}{rrr}\\hfill 3& \\hfill -6& \\hfill 0\\\\ \\hfill 0& \\hfill -3& \\hfill 6\\\\ \\hfill 12& \\hfill 9& \\hfill -18\\end{array}\\right]+\\left[\\begin{array}{rrr}\\hfill -2& \\hfill 4& \\hfill 2\\\\ \\hfill 0& \\hfill -6& \\hfill 4\\\\ \\hfill 0& \\hfill 2& \\hfill -8\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rrr}\\hfill 3 - 2& \\hfill -6+4& \\hfill 0+2\\\\ \\hfill 0+0& \\hfill -3 - 6& \\hfill 6+4\\\\ \\hfill 12+0& \\hfill 9+2& \\hfill -18 - 8\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rrr}\\hfill 1& \\hfill -2& \\hfill 2\\\\ \\hfill 0& \\hfill -9& \\hfill 10\\\\ \\hfill 12& \\hfill 11& \\hfill -26\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm6386\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=6386&theme=oea&iframe_resize_id=ohm6386&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h2 style=\"text-align: left;\">Finding the Product of Two Matrices<\/h2>\n<p>In addition to multiplying a matrix by a scalar, we can multiply two matrices. Finding the <strong>product of two matrices<\/strong> is only possible when the <em>inner dimensions<\/em> 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 [latex]A[\/latex] is an [latex]\\text{ }m\\text{ }\\times \\text{ }r\\text{ }[\/latex] matrix and [latex]B[\/latex] is an [latex]\\text{ }r\\text{ }\\times \\text{ }n\\text{ }[\/latex] matrix, then the product matrix [latex]AB[\/latex] is an [latex]\\text{ }m\\text{ }\\times \\text{ }n\\text{ }[\/latex] matrix. For example, the product [latex]AB[\/latex] is possible because the number of columns in [latex]A[\/latex] is the same as the number of rows in [latex]B[\/latex]. If the inner dimensions do not match, the product is not defined.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03193518\/12.jpg\" alt=\"A has two rows and three columns and B has three rows and three columns. Because the number of columns in A matches the number of rows in B, the product of A and B is defined.\" width=\"154\" height=\"93\" \/><\/p>\n<p>We multiply entries of [latex]A[\/latex] with entries of [latex]B[\/latex] according to a specific pattern as outlined below. The process of <strong>matrix multiplication<\/strong> becomes clearer when working a problem with real numbers.<\/p>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\n<p>Work through the example model of matrix multiplication below on paper, then apply the process to the example problem below. It may take more than once to gain familiarity with it. Don&#8217;t be discouraged if you don&#8217;t understand fully right away. Matrix multiplication is a new skill and it will take time and practice for it to feel comfortable.<\/p>\n<\/div>\n<p>To obtain the entries in row [latex]i[\/latex] of [latex]AB,\\text{}[\/latex] we multiply the entries in row [latex]i[\/latex] of [latex]A[\/latex] by column [latex]j[\/latex] in [latex]B[\/latex] and add. For example, given matrices [latex]A[\/latex] and [latex]B,\\text{}[\/latex] where the dimensions of [latex]A[\/latex] are [latex]2\\text{ }\\times \\text{ }3[\/latex] and the dimensions of [latex]B[\/latex] are [latex]3\\text{ }\\times \\text{ }3,\\text{}[\/latex] the product of [latex]AB[\/latex] will be a [latex]2\\text{ }\\times \\text{ }3[\/latex] matrix.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{rrr}\\hfill {a}_{11}& \\hfill {a}_{12}& \\hfill {a}_{13}\\\\ \\hfill {a}_{21}& \\hfill {a}_{22}& \\hfill {a}_{23}\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{rrr}\\hfill {b}_{11}& \\hfill {b}_{12}& \\hfill {b}_{13}\\\\ \\hfill {b}_{21}& \\hfill {b}_{22}& \\hfill {b}_{23}\\\\ \\hfill {b}_{31}& \\hfill {b}_{32}& \\hfill {b}_{33}\\end{array}\\right][\/latex]<\/p>\n<p>Multiply and add as follows to obtain the first entry of the product matrix [latex]AB[\/latex].<\/p>\n<ol>\n<li>To obtain the entry in row 1, column 1 of [latex]AB,\\text{}[\/latex] multiply the first row in [latex]A[\/latex] by the first column in [latex]B[\/latex] and add.\n<div style=\"text-align: center;\">[latex]\\left[\\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\end{array}\\right]\\cdot \\left[\\begin{array}{c}{b}_{11}\\\\ {b}_{21}\\\\ {b}_{31}\\end{array}\\right]={a}_{11}\\cdot {b}_{11}+{a}_{12}\\cdot {b}_{21}+{a}_{13}\\cdot {b}_{31}[\/latex]<\/div>\n<\/li>\n<li>To obtain the entry in row 1, column 2 of [latex]AB,\\text{}[\/latex] multiply the first row of [latex]A[\/latex] by the second column in [latex]B[\/latex] and add.\n<div style=\"text-align: center;\">[latex]\\left[\\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\end{array}\\right]\\cdot \\left[\\begin{array}{c}{b}_{12}\\\\ {b}_{22}\\\\ {b}_{32}\\end{array}\\right]={a}_{11}\\cdot {b}_{12}+{a}_{12}\\cdot {b}_{22}+{a}_{13}\\cdot {b}_{32}[\/latex]<\/div>\n<\/li>\n<li>To obtain the entry in row 1, column 3 of [latex]AB,\\text{}[\/latex] multiply the first row of [latex]A[\/latex] by the third column in [latex]B[\/latex] and add.\n<div style=\"text-align: center;\">[latex]\\left[\\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\end{array}\\right]\\cdot \\left[\\begin{array}{c}{b}_{13}\\\\ {b}_{23}\\\\ {b}_{33}\\end{array}\\right]={a}_{11}\\cdot {b}_{13}+{a}_{12}\\cdot {b}_{23}+{a}_{13}\\cdot {b}_{33}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p>We proceed the same way to obtain the second row of [latex]AB[\/latex]. In other words, row 2 of [latex]A[\/latex] times column 1 of [latex]B[\/latex]; row 2 of [latex]A[\/latex] times column 2 of [latex]B[\/latex]; row 2 of [latex]A[\/latex] times column 3 of [latex]B[\/latex]. When complete, the product matrix will be<\/p>\n<p style=\"text-align: center;\">[latex]AB=\\left[\\begin{array}{c}\\begin{array}{l}{a}_{11}\\cdot {b}_{11}+{a}_{12}\\cdot {b}_{21}+{a}_{13}\\cdot {b}_{31}\\\\ \\end{array}\\\\ {a}_{21}\\cdot {b}_{11}+{a}_{22}\\cdot {b}_{21}+{a}_{23}\\cdot {b}_{31}\\end{array}\\begin{array}{c}\\begin{array}{l}{a}_{11}\\cdot {b}_{12}+{a}_{12}\\cdot {b}_{22}+{a}_{13}\\cdot {b}_{32}\\\\ \\end{array}\\\\ {a}_{21}\\cdot {b}_{12}+{a}_{22}\\cdot {b}_{22}+{a}_{23}\\cdot {b}_{32}\\end{array}\\begin{array}{c}\\begin{array}{l}{a}_{11}\\cdot {b}_{13}+{a}_{12}\\cdot {b}_{23}+{a}_{13}\\cdot {b}_{33}\\\\ \\end{array}\\\\ {a}_{21}\\cdot {b}_{13}+{a}_{22}\\cdot {b}_{23}+{a}_{23}\\cdot {b}_{33}\\end{array}\\right][\/latex]<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Properties of Matrix Multiplication<\/h3>\n<p>For the matrices [latex]A,B,\\text{}[\/latex] and [latex]C[\/latex] the following properties hold.<\/p>\n<ul>\n<li>Matrix multiplication is associative:\n<div style=\"text-align: center;\">[latex]\\left(AB\\right)C=A\\left(BC\\right)[\/latex]<\/div>\n<\/li>\n<li>Matrix multiplication is distributive:\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ C\\left(A+B\\right)=CA+CB,\\end{array}\\hfill \\\\ \\left(A+B\\right)C=AC+BC.\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<\/ul>\n<p>Note that matrix multiplication is not commutative.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Multiplying Two Matrices<\/h3>\n<p>Multiply matrix [latex]A[\/latex] and matrix [latex]B[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}1& 2\\\\ 3& 4\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{cc}5& 6\\\\ 7& 8\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q843176\">Show Solution<\/span><\/p>\n<div id=\"q843176\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we check the dimensions of the matrices. Matrix [latex]A[\/latex] has dimensions [latex]2\\times 2[\/latex] and matrix [latex]B[\/latex] 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=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03193520\/CNX_Precalc_Figure_09_05_0032.jpg\" alt=\"The first column of the product of A and B is defined as the result of matrix -vector multiplication of A and the first column of B. Column two of the product of A and B is defined as the result of the matrix-vector multiplication of A and the second column of B.\" width=\"487\" height=\"211\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm1075\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1075&theme=oea&iframe_resize_id=ohm1075&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Multiplying Two Matrices<\/h3>\n<p>Given [latex]A[\/latex] and [latex]B:[\/latex]<\/p>\n<ol>\n<li>Find [latex]AB[\/latex].<\/li>\n<li>Find [latex]BA[\/latex].<\/li>\n<\/ol>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}\\hfill -1& \\hfill 2& \\hfill 3\\\\ \\hfill 4& \\hfill 0& \\hfill 5\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{cc}\\hfill 5& \\hfill -1\\\\ \\hfill -4& \\hfill 0\\\\ \\hfill 2& \\hfill 3\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q403957\">Show Solution<\/span><\/p>\n<div id=\"q403957\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>As the dimensions of [latex]A[\/latex] are [latex]2\\text{}\\times \\text{}3[\/latex] and the dimensions of [latex]B[\/latex] are [latex]3\\text{}\\times \\text{}2,\\text{}[\/latex] these matrices can be multiplied together because the number of columns in [latex]A[\/latex] matches the number of rows in [latex]B[\/latex]. The resulting product will be a [latex]2\\text{}\\times \\text{}2[\/latex] matrix, the number of rows in [latex]A[\/latex] by the number of columns in [latex]B[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\hfill \\\\ AB & =\\left[\\begin{array}{rrr}\\hfill -1& \\hfill 2& \\hfill 3\\\\ \\hfill 4& \\hfill 0& \\hfill 5\\end{array}\\right]\\text{ }\\left[\\begin{array}{rr}\\hfill 5& \\hfill -1\\\\ \\hfill -4& \\hfill 0\\\\ \\hfill 2& \\hfill 3\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rr}\\hfill -1\\left(5\\right)+2\\left(-4\\right)+3\\left(2\\right)& \\hfill -1\\left(-1\\right)+2\\left(0\\right)+3\\left(3\\right)\\\\ \\hfill 4\\left(5\\right)+0\\left(-4\\right)+5\\left(2\\right)& \\hfill 4\\left(-1\\right)+0\\left(0\\right)+5\\left(3\\right)\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rr}\\hfill -7& \\hfill 10\\\\ \\hfill 30& \\hfill 11\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<li>The dimensions of [latex]B[\/latex] are [latex]3\\times 2[\/latex] and the dimensions of [latex]A[\/latex] 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.\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\hfill \\\\ BA & =\\left[\\begin{array}{rr}\\hfill 5& \\hfill -1\\\\ \\hfill -4& \\hfill 0\\\\ \\hfill 2& \\hfill 3\\end{array}\\right]\\text{ }\\left[\\begin{array}{rrr}\\hfill -1& \\hfill 2& \\hfill 3\\\\ \\hfill 4& \\hfill 0& \\hfill 5\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rrr}\\hfill 5\\left(-1\\right)+-1\\left(4\\right)& \\hfill 5\\left(2\\right)+-1\\left(0\\right)& \\hfill 5\\left(3\\right)+-1\\left(5\\right)\\\\ \\hfill -4\\left(-1\\right)+0\\left(4\\right)& \\hfill -4\\left(2\\right)+0\\left(0\\right)& \\hfill -4\\left(3\\right)+0\\left(5\\right)\\\\ \\hfill 2\\left(-1\\right)+3\\left(4\\right)& \\hfill 2\\left(2\\right)+3\\left(0\\right)& \\hfill 2\\left(3\\right)+3\\left(5\\right)\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rrr}\\hfill -9& \\hfill 10& \\hfill 10\\\\ \\hfill 4& \\hfill -8& \\hfill -12\\\\ \\hfill 10& \\hfill 4& \\hfill 21\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<h4>Analysis of the Solution<\/h4>\n<p>Notice that the products [latex]AB[\/latex] and [latex]BA[\/latex] are not equal.<\/p>\n<p style=\"text-align: center;\">[latex]AB=\\left[\\begin{array}{cc}-7& 10\\\\ 30& 11\\end{array}\\right]\\ne \\left[\\begin{array}{ccc}-9& 10& 10\\\\ 4& -8& -12\\\\ 10& 4& 21\\end{array}\\right]=BA[\/latex]<\/p>\n<p>This illustrates the fact that matrix multiplication is not commutative.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Is it possible for AB to be defined but not BA?<\/strong><\/p>\n<p><em>Yes, consider a matrix A with dimension [latex]3\\times 4[\/latex] and matrix B with dimension [latex]4\\times 2[\/latex]. 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 class=\"textbox exercises\">\n<h3>Example: Using Matrices in Real-World Problems<\/h3>\n<p>Let\u2019s return to the problem presented at the opening of this section. We have the table below, representing the equipment needs of two soccer teams.<\/p>\n<table summary=\"..\">\n<thead>\n<tr>\n<th><\/th>\n<th style=\"text-align: center;\">Wildcats<\/th>\n<th style=\"text-align: center;\">Mud Cats<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><strong>Goals<\/strong><\/td>\n<td style=\"text-align: center;\">6<\/td>\n<td style=\"text-align: center;\">10<\/td>\n<\/tr>\n<tr>\n<td><strong>Balls<\/strong><\/td>\n<td style=\"text-align: center;\">30<\/td>\n<td style=\"text-align: center;\">24<\/td>\n<\/tr>\n<tr>\n<td><strong>Jerseys<\/strong><\/td>\n<td style=\"text-align: center;\">14<\/td>\n<td style=\"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 the table below.<\/p>\n<table summary=\"..\">\n<tbody>\n<tr>\n<td><strong>Goal<\/strong><\/td>\n<td>$300<\/td>\n<\/tr>\n<tr>\n<td><strong>Ball<\/strong><\/td>\n<td>$10<\/td>\n<\/tr>\n<tr>\n<td><strong>Jersey<\/strong><\/td>\n<td>$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=\\left[\\begin{array}{c}6\\\\ 30\\\\ 14\\end{array}\\begin{array}{c}10\\\\ 24\\\\ 20\\end{array}\\right][\/latex]<\/p>\n<p>The cost matrix is written as<\/p>\n<p style=\"text-align: center;\">[latex]C=\\left[\\begin{array}{ccc}300& 10& 30\\end{array}\\right][\/latex]<br \/>\nWe perform matrix multiplication to obtain costs for the equipment.<br \/>\n[latex]\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ CE & =\\left[\\begin{array}{rrr}\\hfill 300& \\hfill 10& \\hfill 30\\end{array}\\right]\\cdot \\left[\\begin{array}{rr}\\hfill 6& \\hfill 10\\\\ \\hfill 30& \\hfill 24\\\\ \\hfill 14& \\hfill 20\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rr}\\hfill 300\\left(6\\right)+10\\left(30\\right)+30\\left(14\\right)& \\hfill 300\\left(10\\right)+10\\left(24\\right)+30\\left(20\\right)\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rr}\\hfill 2,520& \\hfill 3,840\\end{array}\\right]\\hfill \\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<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a matrix operation, evaluate using a calculator<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Save each matrix as a matrix variable\n<div style=\"text-align: center;\">[latex]\\left[A\\right],\\left[B\\right],\\left[C\\right],..[\/latex]<\/div>\n<\/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 class=\"textbox exercises\">\n<h3>Example: Using a Calculator to Perform Matrix Operations<\/h3>\n<p>Find [latex]AB-C[\/latex]&nbsp;given<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{rrr}\\hfill -15& \\hfill 25& \\hfill 32\\\\ \\hfill 41& \\hfill -7& \\hfill -28\\\\ \\hfill 10& \\hfill 34& \\hfill -2\\end{array}\\right],B=\\left[\\begin{array}{rrr}\\hfill 45& \\hfill 21& \\hfill -37\\\\ \\hfill -24& \\hfill 52& \\hfill 19\\\\ \\hfill 6& \\hfill -48& \\hfill -31\\end{array}\\right],\\text{and }C=\\left[\\begin{array}{rrr}\\hfill -100& \\hfill -89& \\hfill -98\\\\ \\hfill 25& \\hfill -56& \\hfill 74\\\\ \\hfill -67& \\hfill 42& \\hfill -75\\end{array}\\right][\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q32907\">Show Solution<\/span><\/p>\n<div id=\"q32907\" class=\"hidden-answer\" style=\"display: none\">\n<p>On the matrix page of the calculator, we enter matrix [latex]A[\/latex] above as the matrix variable [latex]\\left[A\\right][\/latex], matrix [latex]B[\/latex] above as the matrix variable [latex]\\left[B\\right][\/latex], and matrix [latex]C[\/latex] above as the matrix variable [latex]\\left[C\\right][\/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]\\left[A\\right]\\times \\left[B\\right]-\\left[C\\right][\/latex]<br \/>\nThe calculator gives us the following matrix.<br \/>\n[latex]\\left[\\begin{array}{rrr}\\hfill -983& \\hfill -462& \\hfill 136\\\\ \\hfill 1,820& \\hfill 1,897& \\hfill -856\\\\ \\hfill -311& \\hfill 2,032& \\hfill 413\\end{array}\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-402\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 6386. <strong>Authored by<\/strong>: Dow,David D., mb Sousa,James. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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