{"id":1468,"date":"2025-07-25T02:03:08","date_gmt":"2025-07-25T02:03:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1468"},"modified":"2026-03-24T07:19:40","modified_gmt":"2026-03-24T07:19:40","slug":"matrices-and-matrix-operations-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/matrices-and-matrix-operations-fresh-take\/","title":{"raw":"Matrices and Matrix Operations: Fresh Take","rendered":"Matrices and Matrix Operations: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\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<\/section>\r\n<h2>Matrices<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n\r\nA matrix is a rectangular array of numbers arranged in rows and columns. It's a powerful tool for organizing and manipulating data in various fields, including mathematics, economics, and computer science.\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Matrices organize data in a structured, easy-to-manipulate format<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The size and type of matrix depend on the data being represented<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Element positioning is crucial for correct data interpretation and manipulation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Matrices form the foundation for more advanced linear algebra concepts<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Matrix Basics<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Notation: Matrices are typically enclosed in brackets [ ] or parentheses ( )<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Dimensions: Described as [latex]m \\times n[\/latex], where [latex]m[\/latex] is the number of rows and [latex]n[\/latex] is the number of columns<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Elements: Individual numbers in the matrix, denoted as [latex]a_{ij}[\/latex] (i = row, j = column)<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Types of Matrices<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Square Matrix: Number of rows equals number of columns ([latex]n \\times n[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Row Matrix: Only one row ([latex]1 \\times n[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Column Matrix: Only one column ([latex]m \\times 1[\/latex])<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-abgeecda-ilFJYjfKYjk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ilFJYjfKYjk?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-abgeecda-ilFJYjfKYjk\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851072&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-abgeecda-ilFJYjfKYjk&amp;vembed=0&amp;video_id=ilFJYjfKYjk&amp;video_target=tpm-plugin-abgeecda-ilFJYjfKYjk\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Dimensions+of+a+Matrix_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDimensions of a Matrix\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Adding and Subtracting Matrices<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p class=\"whitespace-pre-wrap break-words\">Matrix addition and subtraction are elementwise operations performed on matrices of the same dimensions.<\/p>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Rules for Matrix Addition and Subtraction<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Matrices must have the same dimensions ([latex]m \\times n[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Add or subtract corresponding elements<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Result is a matrix of the same dimensions<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Formulas<\/strong><\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">For matrices [latex]A[\/latex] and [latex]B[\/latex] with dimensions [latex]m \\times n[\/latex]:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Addition: [latex]C = A + B[\/latex] where [latex]c_{ij} = a_{ij} + b_{ij}[\/latex]\r\nSubtraction: [latex]D = A - B[\/latex] where [latex]d_{ij} = a_{ij} - b_{ij}[\/latex]<\/p>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Properties of Matrix Addition<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Commutative: [latex]A + B = B + A[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Associative: [latex](A + B) + C = A + (B + C)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Additive Identity: [latex]A + 0 = A[\/latex], where [latex]0[\/latex] is a zero matrix<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Additive Inverse: [latex]A + (-A) = 0[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the sum of [latex]A[\/latex] and [latex]B \\text{}[\/latex] given\r\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}a&amp; b\\\\ c&amp; d\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{cc}e&amp; f\\\\ g&amp; h\\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"3634\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"3634\"]\r\n\r\nAdd corresponding entries.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}A+B &amp; =\\left[\\begin{array}{cc}a&amp; b\\\\ c&amp; d\\end{array}\\right]+\\left[\\begin{array}{cc}e&amp; f\\\\ g&amp; h\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{ccc}a+e&amp; &amp; b+f\\\\ c+g&amp; &amp; d+h\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the sum of [latex]A[\/latex] and [latex]B[\/latex].\r\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}4&amp; 1\\\\ 3&amp; 2\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{cc}5&amp; 9\\\\ 0&amp; 7\\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"512431\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"512431\"]\r\n\r\nAdd corresponding entries. Add the entry in row 1, column 1, [latex]{a}_{11},\\text{}[\/latex] of matrix [latex]A[\/latex] to the entry in row 1, column 1, [latex]{b}_{11}[\/latex], of [latex]B[\/latex]. Continue the pattern until all entries have been added.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}A+B &amp; =\\left[\\begin{array}{cc}4&amp; 1\\\\ 3&amp; 2\\end{array}\\right]+\\left[\\begin{array}{cc}5&amp; 9\\\\ 0&amp; 7\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{ccc}4+5&amp; &amp; 1+9\\\\ 3+0&amp; &amp; 2+7\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{cc}9&amp; 10\\\\ 3&amp; 9\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the difference of [latex]A[\/latex] and [latex]B[\/latex].\r\n[latex]A=\\left[\\begin{array}{cc}\\hfill -2&amp; \\hfill 3\\\\ \\hfill 0&amp; \\hfill 1\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{cc}8&amp; 1\\\\ 5&amp; 4\\end{array}\\right][\/latex][reveal-answer q=\"83802\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"83802\"]We subtract the corresponding entries of each matrix.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}A-B &amp; =\\left[\\begin{array}{rr}\\hfill -2&amp; \\hfill 3\\\\ \\hfill 0&amp; \\hfill 1\\end{array}\\right]-\\left[\\begin{array}{rr}\\hfill 8&amp; \\hfill 1\\\\ \\hfill 5&amp; \\hfill 4\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{rrr}\\hfill -2 - 8&amp; \\hfill &amp; \\hfill 3 - 1\\\\ \\hfill 0 - 5&amp; \\hfill &amp; \\hfill 1 - 4\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{rrr}\\hfill -10&amp; \\hfill &amp; \\hfill 2\\\\ \\hfill -5&amp; \\hfill &amp; \\hfill -3\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Add matrix [latex]A[\/latex] and matrix [latex]B[\/latex].[latex]A=\\left[\\begin{array}{rr}\\hfill 2&amp; \\hfill 6\\\\ \\hfill 1&amp; \\hfill 0\\\\ \\hfill 1&amp; \\hfill -3\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{rr}\\hfill 3&amp; \\hfill -2\\\\ \\hfill 1&amp; \\hfill 5\\\\ \\hfill -4&amp; \\hfill 3\\end{array}\\right][\/latex][reveal-answer q=\"644182\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"644182\"][latex]A+B=\\left[\\begin{array}{c}2\\\\ 1\\\\ 1\\end{array}\\begin{array}{c}\\text{ }\\text{ }\\text{ }6\\\\ \\text{ }\\text{ }\\text{ }0\\\\ -3\\end{array}\\right]+\\left[\\begin{array}{c}3\\\\ 1\\\\ -4\\end{array}\\begin{array}{c}-2\\\\ 5\\\\ 3\\end{array}\\right]=\\left[\\begin{array}{c}2+3\\\\ 1+1\\\\ 1+\\left(-4\\right)\\end{array}\\begin{array}{c}6+\\left(-2\\right)\\\\ 0+5\\\\ -3+3\\end{array}\\right]=\\left[\\begin{array}{c}5\\\\ 2\\\\ -3\\end{array}\\begin{array}{c}4\\\\ 5\\\\ 0\\end{array}\\right][\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fgcehcbd-cTKjm5fgqKM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/cTKjm5fgqKM?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fgcehcbd-cTKjm5fgqKM\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851073&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fgcehcbd-cTKjm5fgqKM&amp;vembed=0&amp;video_id=cTKjm5fgqKM&amp;video_target=tpm-plugin-fgcehcbd-cTKjm5fgqKM\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Matrix+Addition+and+Subtraction_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Matrix Addition and Subtraction\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Scalar Multiples of a Matrix<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p class=\"whitespace-pre-wrap break-words\">Scalar multiplication involves multiplying each element of a matrix by a scalar (a real number). This operation is fundamental in matrix algebra and has numerous practical applications.<\/p>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Properties of Scalar Multiplication<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Distributive over matrix addition: [latex]c(A + B) = cA + cB[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Distributive over scalar addition:[latex] (c + d)A = cA + dA[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Associative with scalar multiplication: [latex]c(dA) = (cd)A[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identity: [latex]1A = A[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Given matrix [latex]B,\\text{}[\/latex] find [latex]-2B[\/latex] where\r\n<p style=\"text-align: center;\">[latex]B=\\left[\\begin{array}{cc}4&amp; 1\\\\ 3&amp; 2\\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"2999\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"2999\"][latex]-2B=\\left[\\begin{array}{cc}-8&amp; -2\\\\ -6&amp; -4\\end{array}\\right][\/latex][\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hahhdabf-SYzQPWMUhV8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/SYzQPWMUhV8?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hahhdabf-SYzQPWMUhV8\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851074&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hahhdabf-SYzQPWMUhV8&amp;vembed=0&amp;video_id=SYzQPWMUhV8&amp;video_target=tpm-plugin-hahhdabf-SYzQPWMUhV8\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Matrix+Operations+-+Scalar+Multiplication%2C+Addition%2C+and+Subtraction_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Matrix Operations - Scalar Multiplication, Addition, and Subtraction\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2 style=\"text-align: left;\">Finding the Product of Two Matrices<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Compatibility for Multiplication<\/strong>:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">The number of columns in the first matrix must equal the number of rows in the second matrix.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]A[\/latex] is [latex]m \\times r[\/latex] and [latex]B[\/latex] is [latex]r \\times n[\/latex], then [latex]AB[\/latex] is [latex]m \\times n[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Calculation Process<\/strong>:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">To find each entry in the product matrix:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Multiply each element in a row of the first matrix by the corresponding element in a column of the second matrix.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Add up all these products.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">This process is similar to taking the dot product of a row and a column.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Properties<\/strong>:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Matrix multiplication is associative: [latex]\\left(AB\\right)C=A\\left(BC\\right)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Matrix multiplication is distributive: [latex]C(A+B) = CA + CB[\/latex] and [latex](A+B)C = AC + BC[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Matrix multiplication is NOT commutative: [latex]AB \\neq BA[\/latex] (in general)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Dimensions<\/strong>:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Applications<\/strong>:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Used in various fields including computer graphics, data analysis, and solving systems of linear equations.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Find the product of matrices [latex]A[\/latex] and [latex]B[\/latex]:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\r\nA = \\left[\\begin{array}{cc}\r\n2 &amp; -1 \\\\\r\n3 &amp; 4\r\n\\end{array}\\right], B = \\left[\\begin{array}{cc}\r\n0 &amp; 5 \\\\\r\n-2 &amp; 1\r\n\\end{array}\\right]\r\n[\/latex]<\/p>\r\n[reveal-answer q=\"71906\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"71906\"]\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Check compatibility: [latex]A[\/latex] is 2\u00d72, [latex]B[\/latex] is 2\u00d72, so [latex]AB[\/latex] is defined and will be 2\u00d72.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate each entry:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For the entry in row 1, column 1: [latex](2)(0) + (-1)(-2) = 0 + 2 = 2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For the entry in row 1, column 2: [latex](2)(5) + (-1)(1) = 10 - 1 = 9[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For the entry in row 2, column 1: [latex](3)(0) + (4)(-2) = 0 - 8 = -8[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For the entry in row 2, column 2: [latex](3)(5) + (4)(1) = 15 + 4 = 19[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Write the result: [latex]AB = \\left[\\begin{array}{cc}\r\n2 &amp; 9 \\\\\r\n-8 &amp; 19\r\n\\end{array}\\right][\/latex]<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Find the product of matrices [latex]C[\/latex] and [latex]D[\/latex]:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\r\nC = \\left[\\begin{array}{ccc}\r\n1 &amp; 0 &amp; -2 \\\\\r\n3 &amp; 4 &amp; 1\r\n\\end{array}\\right], D = \\left[\\begin{array}{cc}\r\n2 &amp; -1 \\\\\r\n0 &amp; 3 \\\\\r\n1 &amp; 2\r\n\\end{array}\\right]\r\n[\/latex]<\/p>\r\n[reveal-answer q=\"791271\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"791271\"]\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Check compatibility: [latex]C[\/latex] is 2\u00d73, [latex]D[\/latex] is 3\u00d72, so [latex]CD[\/latex] is defined and will be 2\u00d72.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate each entry:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For the entry in row 1, column 1: [latex](1)(2) + (0)(0) + (-2)(1) = 2 + 0 - 2 = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For the entry in row 1, column 2: [latex](1)(-1) + (0)(3) + (-2)(2) = -1 + 0 - 4 = -5[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For the entry in row 2, column 1: [latex](3)(2) + (4)(0) + (1)(1) = 6 + 0 + 1 = 7[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For the entry in row 2, column 2: [latex](3)(-1) + (4)(3) + (1)(2) = -3 + 12 + 2 = 11[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Write the result: [latex]CD = \\left[\\begin{array}{cc}\r\n0 &amp; -5 \\\\\r\n7 &amp; 11\r\n\\end{array}\\right][\/latex]<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-beaedecf-RqA4HLtJBBs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/RqA4HLtJBBs?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-beaedecf-RqA4HLtJBBs\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851075&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-beaedecf-RqA4HLtJBBs&amp;vembed=0&amp;video_id=RqA4HLtJBBs&amp;video_target=tpm-plugin-beaedecf-RqA4HLtJBBs\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Matrix+Multiplication+(Basic)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Matrix Multiplication (Basic)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\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<\/section>\n<h2>Matrices<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p>A matrix is a rectangular array of numbers arranged in rows and columns. It&#8217;s a powerful tool for organizing and manipulating data in various fields, including mathematics, economics, and computer science.<\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Matrices organize data in a structured, easy-to-manipulate format<\/li>\n<li class=\"whitespace-normal break-words\">The size and type of matrix depend on the data being represented<\/li>\n<li class=\"whitespace-normal break-words\">Element positioning is crucial for correct data interpretation and manipulation<\/li>\n<li class=\"whitespace-normal break-words\">Matrices form the foundation for more advanced linear algebra concepts<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Matrix Basics<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Notation: Matrices are typically enclosed in brackets [ ] or parentheses ( )<\/li>\n<li class=\"whitespace-normal break-words\">Dimensions: Described as [latex]m \\times n[\/latex], where [latex]m[\/latex] is the number of rows and [latex]n[\/latex] is the number of columns<\/li>\n<li class=\"whitespace-normal break-words\">Elements: Individual numbers in the matrix, denoted as [latex]a_{ij}[\/latex] (i = row, j = column)<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>Types of Matrices<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Square Matrix: Number of rows equals number of columns ([latex]n \\times n[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Row Matrix: Only one row ([latex]1 \\times n[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Column Matrix: Only one column ([latex]m \\times 1[\/latex])<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-abgeecda-ilFJYjfKYjk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ilFJYjfKYjk?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-abgeecda-ilFJYjfKYjk\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851072&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-abgeecda-ilFJYjfKYjk&amp;vembed=0&amp;video_id=ilFJYjfKYjk&amp;video_target=tpm-plugin-abgeecda-ilFJYjfKYjk\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Dimensions+of+a+Matrix_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDimensions of a Matrix\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Adding and Subtracting Matrices<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p class=\"whitespace-pre-wrap break-words\">Matrix addition and subtraction are elementwise operations performed on matrices of the same dimensions.<\/p>\n<p class=\"font-600 text-xl font-bold\"><strong>Rules for Matrix Addition and Subtraction<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Matrices must have the same dimensions ([latex]m \\times n[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Add or subtract corresponding elements<\/li>\n<li class=\"whitespace-normal break-words\">Result is a matrix of the same dimensions<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>Formulas<\/strong><\/p>\n<p class=\"whitespace-pre-wrap break-words\">For matrices [latex]A[\/latex] and [latex]B[\/latex] with dimensions [latex]m \\times n[\/latex]:<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Addition: [latex]C = A + B[\/latex] where [latex]c_{ij} = a_{ij} + b_{ij}[\/latex]<br \/>\nSubtraction: [latex]D = A - B[\/latex] where [latex]d_{ij} = a_{ij} - b_{ij}[\/latex]<\/p>\n<p class=\"font-600 text-xl font-bold\"><strong>Properties of Matrix Addition<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Commutative: [latex]A + B = B + A[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Associative: [latex](A + B) + C = A + (B + C)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Additive Identity: [latex]A + 0 = A[\/latex], where [latex]0[\/latex] is a zero matrix<\/li>\n<li class=\"whitespace-normal break-words\">Additive Inverse: [latex]A + (-A) = 0[\/latex]<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the sum of [latex]A[\/latex] and [latex]B \\text{}[\/latex] given<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}a& b\\\\ c& d\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{cc}e& f\\\\ g& h\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q3634\">Show Solution<\/button><\/p>\n<div id=\"q3634\" class=\"hidden-answer\" style=\"display: none\">\n<p>Add corresponding entries.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}A+B & =\\left[\\begin{array}{cc}a& b\\\\ c& d\\end{array}\\right]+\\left[\\begin{array}{cc}e& f\\\\ g& h\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{ccc}a+e& & b+f\\\\ c+g& & d+h\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the sum of [latex]A[\/latex] and [latex]B[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}4& 1\\\\ 3& 2\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{cc}5& 9\\\\ 0& 7\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q512431\">Show Solution<\/button><\/p>\n<div id=\"q512431\" class=\"hidden-answer\" style=\"display: none\">\n<p>Add corresponding entries. Add the entry in row 1, column 1, [latex]{a}_{11},\\text{}[\/latex] of matrix [latex]A[\/latex] to the entry in row 1, column 1, [latex]{b}_{11}[\/latex], of [latex]B[\/latex]. Continue the pattern until all entries have been added.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}A+B & =\\left[\\begin{array}{cc}4& 1\\\\ 3& 2\\end{array}\\right]+\\left[\\begin{array}{cc}5& 9\\\\ 0& 7\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{ccc}4+5& & 1+9\\\\ 3+0& & 2+7\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{cc}9& 10\\\\ 3& 9\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the difference of [latex]A[\/latex] and [latex]B[\/latex].<br \/>\n[latex]A=\\left[\\begin{array}{cc}\\hfill -2& \\hfill 3\\\\ \\hfill 0& \\hfill 1\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{cc}8& 1\\\\ 5& 4\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q83802\">Show Solution<\/button><\/p>\n<div id=\"q83802\" class=\"hidden-answer\" style=\"display: none\">We subtract the corresponding entries of each matrix.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}A-B & =\\left[\\begin{array}{rr}\\hfill -2& \\hfill 3\\\\ \\hfill 0& \\hfill 1\\end{array}\\right]-\\left[\\begin{array}{rr}\\hfill 8& \\hfill 1\\\\ \\hfill 5& \\hfill 4\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rrr}\\hfill -2 - 8& \\hfill & \\hfill 3 - 1\\\\ \\hfill 0 - 5& \\hfill & \\hfill 1 - 4\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rrr}\\hfill -10& \\hfill & \\hfill 2\\\\ \\hfill -5& \\hfill & \\hfill -3\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Add matrix [latex]A[\/latex] and matrix [latex]B[\/latex].[latex]A=\\left[\\begin{array}{rr}\\hfill 2& \\hfill 6\\\\ \\hfill 1& \\hfill 0\\\\ \\hfill 1& \\hfill -3\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{rr}\\hfill 3& \\hfill -2\\\\ \\hfill 1& \\hfill 5\\\\ \\hfill -4& \\hfill 3\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q644182\">Show Solution<\/button><\/p>\n<div id=\"q644182\" class=\"hidden-answer\" style=\"display: none\">[latex]A+B=\\left[\\begin{array}{c}2\\\\ 1\\\\ 1\\end{array}\\begin{array}{c}\\text{ }\\text{ }\\text{ }6\\\\ \\text{ }\\text{ }\\text{ }0\\\\ -3\\end{array}\\right]+\\left[\\begin{array}{c}3\\\\ 1\\\\ -4\\end{array}\\begin{array}{c}-2\\\\ 5\\\\ 3\\end{array}\\right]=\\left[\\begin{array}{c}2+3\\\\ 1+1\\\\ 1+\\left(-4\\right)\\end{array}\\begin{array}{c}6+\\left(-2\\right)\\\\ 0+5\\\\ -3+3\\end{array}\\right]=\\left[\\begin{array}{c}5\\\\ 2\\\\ -3\\end{array}\\begin{array}{c}4\\\\ 5\\\\ 0\\end{array}\\right][\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fgcehcbd-cTKjm5fgqKM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/cTKjm5fgqKM?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fgcehcbd-cTKjm5fgqKM\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851073&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fgcehcbd-cTKjm5fgqKM&amp;vembed=0&amp;video_id=cTKjm5fgqKM&amp;video_target=tpm-plugin-fgcehcbd-cTKjm5fgqKM\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Matrix+Addition+and+Subtraction_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Matrix Addition and Subtraction\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Scalar Multiples of a Matrix<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p class=\"whitespace-pre-wrap break-words\">Scalar multiplication involves multiplying each element of a matrix by a scalar (a real number). This operation is fundamental in matrix algebra and has numerous practical applications.<\/p>\n<p class=\"font-600 text-xl font-bold\"><strong>Properties of Scalar Multiplication<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Distributive over matrix addition: [latex]c(A + B) = cA + cB[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Distributive over scalar addition:[latex](c + d)A = cA + dA[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Associative with scalar multiplication: [latex]c(dA) = (cd)A[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Identity: [latex]1A = A[\/latex]<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q2999\">Show Solution<\/button><\/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<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hahhdabf-SYzQPWMUhV8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/SYzQPWMUhV8?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hahhdabf-SYzQPWMUhV8\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851074&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hahhdabf-SYzQPWMUhV8&amp;vembed=0&amp;video_id=SYzQPWMUhV8&amp;video_target=tpm-plugin-hahhdabf-SYzQPWMUhV8\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Matrix+Operations+-+Scalar+Multiplication%2C+Addition%2C+and+Subtraction_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Matrix Operations &#8211; Scalar Multiplication, Addition, and Subtraction\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2 style=\"text-align: left;\">Finding the Product of Two Matrices<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\"><strong>Compatibility for Multiplication<\/strong>:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The number of columns in the first matrix must equal the number of rows in the second matrix.<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]A[\/latex] is [latex]m \\times r[\/latex] and [latex]B[\/latex] is [latex]r \\times n[\/latex], then [latex]AB[\/latex] is [latex]m \\times n[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Calculation Process<\/strong>:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">To find each entry in the product matrix:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Multiply each element in a row of the first matrix by the corresponding element in a column of the second matrix.<\/li>\n<li class=\"whitespace-normal break-words\">Add up all these products.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">This process is similar to taking the dot product of a row and a column.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Properties<\/strong>:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Matrix multiplication is associative: [latex]\\left(AB\\right)C=A\\left(BC\\right)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Matrix multiplication is distributive: [latex]C(A+B) = CA + CB[\/latex] and [latex](A+B)C = AC + BC[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Matrix multiplication is NOT commutative: [latex]AB \\neq BA[\/latex] (in general)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Dimensions<\/strong>:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Applications<\/strong>:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Used in various fields including computer graphics, data analysis, and solving systems of linear equations.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Find the product of matrices [latex]A[\/latex] and [latex]B[\/latex]:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]A = \\left[\\begin{array}{cc}  2 & -1 \\\\  3 & 4  \\end{array}\\right], B = \\left[\\begin{array}{cc}  0 & 5 \\\\  -2 & 1  \\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q71906\">Show Answer<\/button><\/p>\n<div id=\"q71906\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li class=\"whitespace-normal break-words\">Check compatibility: [latex]A[\/latex] is 2\u00d72, [latex]B[\/latex] is 2\u00d72, so [latex]AB[\/latex] is defined and will be 2\u00d72.<\/li>\n<li class=\"whitespace-normal break-words\">Calculate each entry:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For the entry in row 1, column 1: [latex](2)(0) + (-1)(-2) = 0 + 2 = 2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">For the entry in row 1, column 2: [latex](2)(5) + (-1)(1) = 10 - 1 = 9[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">For the entry in row 2, column 1: [latex](3)(0) + (4)(-2) = 0 - 8 = -8[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">For the entry in row 2, column 2: [latex](3)(5) + (4)(1) = 15 + 4 = 19[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Write the result: [latex]AB = \\left[\\begin{array}{cc}  2 & 9 \\\\  -8 & 19  \\end{array}\\right][\/latex]<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Find the product of matrices [latex]C[\/latex] and [latex]D[\/latex]:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]C = \\left[\\begin{array}{ccc}  1 & 0 & -2 \\\\  3 & 4 & 1  \\end{array}\\right], D = \\left[\\begin{array}{cc}  2 & -1 \\\\  0 & 3 \\\\  1 & 2  \\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q791271\">Show Answer<\/button><\/p>\n<div id=\"q791271\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li class=\"whitespace-normal break-words\">Check compatibility: [latex]C[\/latex] is 2\u00d73, [latex]D[\/latex] is 3\u00d72, so [latex]CD[\/latex] is defined and will be 2\u00d72.<\/li>\n<li class=\"whitespace-normal break-words\">Calculate each entry:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For the entry in row 1, column 1: [latex](1)(2) + (0)(0) + (-2)(1) = 2 + 0 - 2 = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">For the entry in row 1, column 2: [latex](1)(-1) + (0)(3) + (-2)(2) = -1 + 0 - 4 = -5[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">For the entry in row 2, column 1: [latex](3)(2) + (4)(0) + (1)(1) = 6 + 0 + 1 = 7[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">For the entry in row 2, column 2: [latex](3)(-1) + (4)(3) + (1)(2) = -3 + 12 + 2 = 11[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Write the result: [latex]CD = \\left[\\begin{array}{cc}  0 & -5 \\\\  7 & 11  \\end{array}\\right][\/latex]<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-beaedecf-RqA4HLtJBBs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/RqA4HLtJBBs?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-beaedecf-RqA4HLtJBBs\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851075&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-beaedecf-RqA4HLtJBBs&amp;vembed=0&amp;video_id=RqA4HLtJBBs&amp;video_target=tpm-plugin-beaedecf-RqA4HLtJBBs\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Matrix+Multiplication+(Basic)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Matrix Multiplication (Basic)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Dimensions of a Matrix\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/ilFJYjfKYjk\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Matrix Addition and Subtraction\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/cTKjm5fgqKM\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Matrix Operations - 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