{"id":1474,"date":"2025-07-25T02:05:27","date_gmt":"2025-07-25T02:05:27","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1474"},"modified":"2026-03-24T07:16:46","modified_gmt":"2026-03-24T07:16:46","slug":"solving-systems-with-inverses-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/solving-systems-with-inverses-fresh-take\/","title":{"raw":"Solving Systems with Inverses: Fresh Take","rendered":"Solving Systems with Inverses: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find the inverse of a matrix.<\/li>\r\n \t<li>Solve a system of linear equations using an inverse matrix.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Finding the Inverse of a Matrix<\/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\">Matrix Inverses:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For a square matrix [latex]A[\/latex], its inverse [latex]A^{-1}[\/latex] satisfies: [latex]AA^{-1} = A^{-1}A = I[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]I[\/latex] is the identity matrix, with [latex]1[\/latex]s on the main diagonal and [latex]0[\/latex]s elsewhere<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Not all square matrices have inverses; those that do are called invertible<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Properties of Matrix Inverses:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Only square matrices can have inverses<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]A[\/latex] is invertible, its inverse [latex]A^{-1}[\/latex] is unique<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The identity matrix acts like [latex]1[\/latex] in matrix algebra: [latex]AI = IA = A[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identity Matrix:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">2x2 Identity Matrix: [latex]I_2 = \\begin{bmatrix} 1 &amp; 0 \\ 0 &amp; 1 \\end{bmatrix}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">3x3 Identity Matrix: [latex]I_3 = \\begin{bmatrix} 1 &amp; 0 &amp; 0 \\ 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1 \\end{bmatrix}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solving Systems of Equations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For a system [latex]AX = B[\/latex], the solution is [latex]X = A^{-1}B[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">This method is efficient for larger systems of equations<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Verifying Inverses:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">To show that [latex]B[\/latex] is the inverse of [latex]A[\/latex], prove that [latex]AB = BA = I[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Show that the following two matrices are inverses of each other.\r\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{rrr}\\hfill 1&amp; \\hfill &amp; \\hfill 4\\\\ \\hfill -1&amp; \\hfill &amp; \\hfill -3\\end{array}\\right],B=\\left[\\begin{array}{rrr}\\hfill -3&amp; \\hfill &amp; \\hfill -4\\\\ \\hfill 1&amp; \\hfill &amp; \\hfill 1\\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"159815\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"159815\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}AB=\\left[\\begin{array}{rrr}\\hfill 1&amp; \\hfill &amp; \\hfill 4\\\\ \\hfill -1&amp; \\hfill &amp; \\hfill -3\\end{array}\\right]\\begin{array}{r}\\hfill \\end{array}\\left[\\begin{array}{rrr}\\hfill -3&amp; \\hfill &amp; \\hfill -4\\\\ \\hfill 1&amp; \\hfill &amp; \\hfill 1\\end{array}\\right]=\\left[\\begin{array}{rrr}\\hfill 1\\left(-3\\right)+4\\left(1\\right)&amp; \\hfill &amp; \\hfill 1\\left(-4\\right)+4\\left(1\\right)\\\\ \\hfill -1\\left(-3\\right)+-3\\left(1\\right)&amp; \\hfill &amp; \\hfill -1\\left(-4\\right)+-3\\left(1\\right)\\end{array}\\right]=\\left[\\begin{array}{rrr}\\hfill 1&amp; \\hfill &amp; \\hfill 0\\\\ \\hfill 0&amp; \\hfill &amp; \\hfill 1\\end{array}\\right]\\hfill \\\\ BA=\\left[\\begin{array}{rrr}\\hfill -3&amp; \\hfill &amp; \\hfill -4\\\\ \\hfill 1&amp; \\hfill &amp; \\hfill 1\\end{array}\\right]\\begin{array}{r}\\hfill \\end{array}\\left[\\begin{array}{rrr}\\hfill 1&amp; \\hfill &amp; \\hfill 4\\\\ \\hfill -1&amp; \\hfill &amp; \\hfill -3\\end{array}\\right]=\\left[\\begin{array}{rrr}\\hfill -3\\left(1\\right)+-4\\left(-1\\right)&amp; \\hfill &amp; \\hfill -3\\left(4\\right)+-4\\left(-3\\right)\\\\ \\hfill 1\\left(1\\right)+1\\left(-1\\right)&amp; \\hfill &amp; \\hfill 1\\left(4\\right)+1\\left(-3\\right)\\end{array}\\right]=\\left[\\begin{array}{rrr}\\hfill 1&amp; \\hfill &amp; \\hfill 0\\\\ \\hfill 0&amp; \\hfill &amp; \\hfill 1\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\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-ebdebbaa-hPAS6H6xFa0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/hPAS6H6xFa0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ebdebbaa-hPAS6H6xFa0\" 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=12851174&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ebdebbaa-hPAS6H6xFa0&amp;vembed=0&amp;video_id=hPAS6H6xFa0&amp;video_target=tpm-plugin-ebdebbaa-hPAS6H6xFa0\" 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\/The+Identity+Matrix_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cThe Identity Matrix\u201d here (opens in new window).<\/a>\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-ghfbfhdc-KBYvP6YG58g\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/KBYvP6YG58g?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ghfbfhdc-KBYvP6YG58g\" 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=12851175&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ghfbfhdc-KBYvP6YG58g&amp;vembed=0&amp;video_id=KBYvP6YG58g&amp;video_target=tpm-plugin-ghfbfhdc-KBYvP6YG58g\" 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\/Determining+Inverse+Matrices+Using+Augmented+Matrices_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermining Inverse Matrices Using Augmented Matrices\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>&nbsp;\r\n<h2>Finding the Multiplicative Inverse<\/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\">Definition of Matrix Inverse:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For a square matrix [latex]A[\/latex], its inverse [latex]A^{-1}[\/latex] satisfies: [latex]AA^{-1} = A^{-1}A = I[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]I[\/latex] is the identity matrix<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Methods to Find Matrix Inverses:\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Matrix Multiplication Method<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Augmenting with Identity Method<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Inverse Formula (for 2x2 matrices)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Matrix Multiplication Method:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Set up equation: [latex]A \\begin{bmatrix} a &amp; b \\ c &amp; d \\end{bmatrix} = \\begin{bmatrix} 1 &amp; 0 \\ 0 &amp; 1 \\end{bmatrix}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve system of equations to find [latex]a, b, c, d[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Augmenting with Identity Method:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Create augmented matrix [latex][A|I][\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Perform row operations to transform [latex]A[\/latex] into [latex]I[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The transformed I becomes [latex]A^{-1}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Inverse Formula for 2x2 Matrices:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For [latex]A = \\begin{bmatrix} a &amp; b \\ c &amp; d \\end{bmatrix}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]A^{-1} = \\frac{1}{ad-bc} \\begin{bmatrix} d &amp; -b \\ -c &amp; a \\end{bmatrix}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Only valid if [latex]ad-bc \\neq 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Use the formula to find the inverse of matrix [latex]A[\/latex]. Verify your answer by augmenting with the identity matrix.\r\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc} \\hfill 1&amp; \\hfill -1\\\\ \\hfill 2&amp; \\hfill 3\\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"161972\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"161972\"]\r\n\r\n[latex]{A}^{-1}=\\left[\\begin{array}{cc} \\hfill \\frac{3}{5}&amp; \\hfill \\frac{1}{5}\\\\ \\hfill -\\frac{2}{5}&amp; \\hfill \\frac{1}{5}\\end{array}\\right][\/latex][\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the inverse, if it exists, of the given matrix.\r\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}3&amp; 6\\\\ 1&amp; 2\\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"930665\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"930665\"]\r\n\r\nWe will use the method of augmenting with the identity.\r\n<p style=\"text-align: center;\">[latex]\\left[\\begin{array}{cc|cc}\\hfill 3&amp; \\hfill 6&amp; \\hfill 1&amp; \\hfill 0\\\\ \\hfill 1&amp; \\hfill 2&amp; \\hfill 0&amp; \\hfill 1\\\\ \\end{array}\\right][\/latex]<\/p>\r\n\r\n<ol>\r\n \t<li>Switch row 1 and row 2.\r\n[latex]\\left[\\begin{array}{cc|cc}\\hfill 1&amp; \\hfill 2&amp; \\hfill 0&amp; \\hfill 1\\\\ \\hfill 3&amp; \\hfill 6&amp; \\hfill 1&amp; \\hfill 0\\\\ \\end{array}\\right][\/latex]<\/li>\r\n \t<li>Multiply row 1 by \u22123 and add it to row 2.\r\n[latex]\\left[\\begin{array}{cc|cc}\\hfill 1&amp; \\hfill 2&amp; \\hfill 1&amp; \\hfill 0\\\\ \\hfill 0&amp; \\hfill 0&amp; \\hfill 1&amp; \\hfill -3\\\\ \\end{array}\\right][\/latex]<\/li>\r\n \t<li>There is nothing further we can do. The zeros in row 2 indicate that this matrix has no inverse.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Multiplicative Inverse of 3\u00d73 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\">Definition of Matrix Inverse:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For a square matrix [latex]A[\/latex], its inverse [latex]A^{-1}[\/latex] satisfies: [latex]AA^{-1} = A^{-1}A = I[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]I[\/latex] is the 3x3 identity matrix: [latex]I = \\begin{bmatrix} 1 &amp; 0 &amp; 0 \\ 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1 \\end{bmatrix}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Method for Finding 3x3 Matrix Inverse:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Augment the original matrix with the identity matrix<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Perform row operations to transform the left side into the identity matrix<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The right side becomes the inverse matrix<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Steps to Find 3x3 Matrix Inverse:\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Write the augmented matrix [latex][A|I][\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use elementary row operations to transform [latex]A[\/latex] into [latex]I[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The resulting right side is [latex]A^{-1}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Singularity:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Not all 3x3 matrices have inverses<\/li>\r\n \t<li class=\"whitespace-normal break-words\">A matrix with no inverse is called singular<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If row operations result in a row of zeros, the matrix is singular<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Verification:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Check that [latex]AA^{-1} = A^{-1}A = I[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the inverse of the [latex]3\\times 3[\/latex] matrix.\r\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}\\hfill 2&amp;\\hfill -17&amp; \\hfill 11\\\\ \\hfill -1&amp; \\hfill 11&amp; \\hfill -7\\\\ \\hfill 0&amp; \\hfill 3&amp; \\hfill -2\\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"202597\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"202597\"]\r\n\r\n[latex]{A}^{-1}=\\left[\\begin{array}{ccc}\\hfill 1&amp; \\hfill 1&amp; \\hfill 2\\\\ \\hfill 2&amp; \\hfill 4&amp; \\hfill -3\\\\ \\hfill 3&amp; \\hfill 6&amp; \\hfill -5\\end{array}\\right][\/latex][\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Solving a System of Linear Equations Using the Inverse of a Matrix<\/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\">Matrix Representation of Systems:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]AX = B[\/latex], where:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]A[\/latex]: Coefficient matrix<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]X[\/latex]: Variable matrix<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]B[\/latex]: Constant matrix<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solution Method:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Multiply both sides by [latex]A^{-1}[\/latex]: [latex]A^{-1}AX = A^{-1}B[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify: [latex]IX = A^{-1}B[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solution: [latex]X = A^{-1}B[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Matrix Inverse Properties:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]A^{-1}A = AA^{-1} = I[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Not all matrices have inverses (singular matrices)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Order Matters:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]A^{-1}B \\neq BA^{-1}[\/latex] (matrix multiplication is not commutative)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Solve the system using the inverse of the coefficient matrix.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{ }2x - 17y+11z=0\\hfill \\\\ \\text{ }-x+11y - 7z=8\\hfill \\\\ \\text{ }3y - 2z=-2\\hfill \\end{array}[\/latex]<\/p>\r\n[reveal-answer q=\"514137\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"514137\"]\r\n\r\n[latex]X=\\left[\\begin{array}{c}4\\\\ 38\\\\ 58\\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-egbgcebh-ieFpNMrd9kU\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ieFpNMrd9kU?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-egbgcebh-ieFpNMrd9kU\" 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=12851176&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-egbgcebh-ieFpNMrd9kU&amp;vembed=0&amp;video_id=ieFpNMrd9kU&amp;video_target=tpm-plugin-egbgcebh-ieFpNMrd9kU\" 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\/Using+a+Matrix+Equation+to+Solve+a+System+of+Equations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cUsing a Matrix Equation to Solve a System of Equations\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 inverse of a matrix.<\/li>\n<li>Solve a system of linear equations using an inverse matrix.<\/li>\n<\/ul>\n<\/section>\n<h2>Finding the Inverse of a Matrix<\/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\">Matrix Inverses:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For a square matrix [latex]A[\/latex], its inverse [latex]A^{-1}[\/latex] satisfies: [latex]AA^{-1} = A^{-1}A = I[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]I[\/latex] is the identity matrix, with [latex]1[\/latex]s on the main diagonal and [latex]0[\/latex]s elsewhere<\/li>\n<li class=\"whitespace-normal break-words\">Not all square matrices have inverses; those that do are called invertible<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Properties of Matrix Inverses:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Only square matrices can have inverses<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]A[\/latex] is invertible, its inverse [latex]A^{-1}[\/latex] is unique<\/li>\n<li class=\"whitespace-normal break-words\">The identity matrix acts like [latex]1[\/latex] in matrix algebra: [latex]AI = IA = A[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Identity Matrix:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">2&#215;2 Identity Matrix: [latex]I_2 = \\begin{bmatrix} 1 & 0 \\ 0 & 1 \\end{bmatrix}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">3&#215;3 Identity Matrix: [latex]I_3 = \\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\end{bmatrix}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solving Systems of Equations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For a system [latex]AX = B[\/latex], the solution is [latex]X = A^{-1}B[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">This method is efficient for larger systems of equations<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Verifying Inverses:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">To show that [latex]B[\/latex] is the inverse of [latex]A[\/latex], prove that [latex]AB = BA = I[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Show that the following two matrices are inverses of each other.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{rrr}\\hfill 1& \\hfill & \\hfill 4\\\\ \\hfill -1& \\hfill & \\hfill -3\\end{array}\\right],B=\\left[\\begin{array}{rrr}\\hfill -3& \\hfill & \\hfill -4\\\\ \\hfill 1& \\hfill & \\hfill 1\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q159815\">Show Solution<\/button><\/p>\n<div id=\"q159815\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}AB=\\left[\\begin{array}{rrr}\\hfill 1& \\hfill & \\hfill 4\\\\ \\hfill -1& \\hfill & \\hfill -3\\end{array}\\right]\\begin{array}{r}\\hfill \\end{array}\\left[\\begin{array}{rrr}\\hfill -3& \\hfill & \\hfill -4\\\\ \\hfill 1& \\hfill & \\hfill 1\\end{array}\\right]=\\left[\\begin{array}{rrr}\\hfill 1\\left(-3\\right)+4\\left(1\\right)& \\hfill & \\hfill 1\\left(-4\\right)+4\\left(1\\right)\\\\ \\hfill -1\\left(-3\\right)+-3\\left(1\\right)& \\hfill & \\hfill -1\\left(-4\\right)+-3\\left(1\\right)\\end{array}\\right]=\\left[\\begin{array}{rrr}\\hfill 1& \\hfill & \\hfill 0\\\\ \\hfill 0& \\hfill & \\hfill 1\\end{array}\\right]\\hfill \\\\ BA=\\left[\\begin{array}{rrr}\\hfill -3& \\hfill & \\hfill -4\\\\ \\hfill 1& \\hfill & \\hfill 1\\end{array}\\right]\\begin{array}{r}\\hfill \\end{array}\\left[\\begin{array}{rrr}\\hfill 1& \\hfill & \\hfill 4\\\\ \\hfill -1& \\hfill & \\hfill -3\\end{array}\\right]=\\left[\\begin{array}{rrr}\\hfill -3\\left(1\\right)+-4\\left(-1\\right)& \\hfill & \\hfill -3\\left(4\\right)+-4\\left(-3\\right)\\\\ \\hfill 1\\left(1\\right)+1\\left(-1\\right)& \\hfill & \\hfill 1\\left(4\\right)+1\\left(-3\\right)\\end{array}\\right]=\\left[\\begin{array}{rrr}\\hfill 1& \\hfill & \\hfill 0\\\\ \\hfill 0& \\hfill & \\hfill 1\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\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-ebdebbaa-hPAS6H6xFa0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/hPAS6H6xFa0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ebdebbaa-hPAS6H6xFa0\" 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=12851174&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ebdebbaa-hPAS6H6xFa0&amp;vembed=0&amp;video_id=hPAS6H6xFa0&amp;video_target=tpm-plugin-ebdebbaa-hPAS6H6xFa0\" 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\/The+Identity+Matrix_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cThe Identity Matrix\u201d here (opens in new window).<\/a><\/p>\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-ghfbfhdc-KBYvP6YG58g\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/KBYvP6YG58g?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ghfbfhdc-KBYvP6YG58g\" 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=12851175&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ghfbfhdc-KBYvP6YG58g&amp;vembed=0&amp;video_id=KBYvP6YG58g&amp;video_target=tpm-plugin-ghfbfhdc-KBYvP6YG58g\" 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\/Determining+Inverse+Matrices+Using+Augmented+Matrices_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermining Inverse Matrices Using Augmented Matrices\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>&nbsp;<\/p>\n<h2>Finding the Multiplicative Inverse<\/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\">Definition of Matrix Inverse:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For a square matrix [latex]A[\/latex], its inverse [latex]A^{-1}[\/latex] satisfies: [latex]AA^{-1} = A^{-1}A = I[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]I[\/latex] is the identity matrix<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Methods to Find Matrix Inverses:\n<ul>\n<li class=\"whitespace-normal break-words\">Matrix Multiplication Method<\/li>\n<li class=\"whitespace-normal break-words\">Augmenting with Identity Method<\/li>\n<li class=\"whitespace-normal break-words\">Inverse Formula (for 2&#215;2 matrices)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Matrix Multiplication Method:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Set up equation: [latex]A \\begin{bmatrix} a & b \\ c & d \\end{bmatrix} = \\begin{bmatrix} 1 & 0 \\ 0 & 1 \\end{bmatrix}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Solve system of equations to find [latex]a, b, c, d[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Augmenting with Identity Method:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Create augmented matrix [latex][A|I][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Perform row operations to transform [latex]A[\/latex] into [latex]I[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">The transformed I becomes [latex]A^{-1}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Inverse Formula for 2&#215;2 Matrices:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]A = \\begin{bmatrix} a & b \\ c & d \\end{bmatrix}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]A^{-1} = \\frac{1}{ad-bc} \\begin{bmatrix} d & -b \\ -c & a \\end{bmatrix}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Only valid if [latex]ad-bc \\neq 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Use the formula to find the inverse of matrix [latex]A[\/latex]. Verify your answer by augmenting with the identity matrix.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc} \\hfill 1& \\hfill -1\\\\ \\hfill 2& \\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=\"q161972\">Show Solution<\/button><\/p>\n<div id=\"q161972\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{A}^{-1}=\\left[\\begin{array}{cc} \\hfill \\frac{3}{5}& \\hfill \\frac{1}{5}\\\\ \\hfill -\\frac{2}{5}& \\hfill \\frac{1}{5}\\end{array}\\right][\/latex]<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the inverse, if it exists, of the given matrix.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}3& 6\\\\ 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=\"q930665\">Show Solution<\/button><\/p>\n<div id=\"q930665\" class=\"hidden-answer\" style=\"display: none\">\n<p>We will use the method of augmenting with the identity.<\/p>\n<p style=\"text-align: center;\">[latex]\\left[\\begin{array}{cc|cc}\\hfill 3& \\hfill 6& \\hfill 1& \\hfill 0\\\\ \\hfill 1& \\hfill 2& \\hfill 0& \\hfill 1\\\\ \\end{array}\\right][\/latex]<\/p>\n<ol>\n<li>Switch row 1 and row 2.<br \/>\n[latex]\\left[\\begin{array}{cc|cc}\\hfill 1& \\hfill 2& \\hfill 0& \\hfill 1\\\\ \\hfill 3& \\hfill 6& \\hfill 1& \\hfill 0\\\\ \\end{array}\\right][\/latex]<\/li>\n<li>Multiply row 1 by \u22123 and add it to row 2.<br \/>\n[latex]\\left[\\begin{array}{cc|cc}\\hfill 1& \\hfill 2& \\hfill 1& \\hfill 0\\\\ \\hfill 0& \\hfill 0& \\hfill 1& \\hfill -3\\\\ \\end{array}\\right][\/latex]<\/li>\n<li>There is nothing further we can do. The zeros in row 2 indicate that this matrix has no inverse.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<h2>Multiplicative Inverse of 3\u00d73 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\">Definition of Matrix Inverse:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For a square matrix [latex]A[\/latex], its inverse [latex]A^{-1}[\/latex] satisfies: [latex]AA^{-1} = A^{-1}A = I[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]I[\/latex] is the 3&#215;3 identity matrix: [latex]I = \\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\end{bmatrix}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Method for Finding 3&#215;3 Matrix Inverse:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Augment the original matrix with the identity matrix<\/li>\n<li class=\"whitespace-normal break-words\">Perform row operations to transform the left side into the identity matrix<\/li>\n<li class=\"whitespace-normal break-words\">The right side becomes the inverse matrix<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Steps to Find 3&#215;3 Matrix Inverse:\n<ul>\n<li class=\"whitespace-normal break-words\">Write the augmented matrix [latex][A|I][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Use elementary row operations to transform [latex]A[\/latex] into [latex]I[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">The resulting right side is [latex]A^{-1}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Singularity:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Not all 3&#215;3 matrices have inverses<\/li>\n<li class=\"whitespace-normal break-words\">A matrix with no inverse is called singular<\/li>\n<li class=\"whitespace-normal break-words\">If row operations result in a row of zeros, the matrix is singular<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Verification:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Check that [latex]AA^{-1} = A^{-1}A = I[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the inverse of the [latex]3\\times 3[\/latex] matrix.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}\\hfill 2&\\hfill -17& \\hfill 11\\\\ \\hfill -1& \\hfill 11& \\hfill -7\\\\ \\hfill 0& \\hfill 3& \\hfill -2\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q202597\">Show Solution<\/button><\/p>\n<div id=\"q202597\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{A}^{-1}=\\left[\\begin{array}{ccc}\\hfill 1& \\hfill 1& \\hfill 2\\\\ \\hfill 2& \\hfill 4& \\hfill -3\\\\ \\hfill 3& \\hfill 6& \\hfill -5\\end{array}\\right][\/latex]<\/p><\/div>\n<\/div>\n<\/section>\n<h2>Solving a System of Linear Equations Using the Inverse of a Matrix<\/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\">Matrix Representation of Systems:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]AX = B[\/latex], where:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]A[\/latex]: Coefficient matrix<\/li>\n<li class=\"whitespace-normal break-words\">[latex]X[\/latex]: Variable matrix<\/li>\n<li class=\"whitespace-normal break-words\">[latex]B[\/latex]: Constant matrix<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solution Method:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Multiply both sides by [latex]A^{-1}[\/latex]: [latex]A^{-1}AX = A^{-1}B[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Simplify: [latex]IX = A^{-1}B[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Solution: [latex]X = A^{-1}B[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Matrix Inverse Properties:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]A^{-1}A = AA^{-1} = I[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Not all matrices have inverses (singular matrices)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Order Matters:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]A^{-1}B \\neq BA^{-1}[\/latex] (matrix multiplication is not commutative)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the system using the inverse of the coefficient matrix.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{ }2x - 17y+11z=0\\hfill \\\\ \\text{ }-x+11y - 7z=8\\hfill \\\\ \\text{ }3y - 2z=-2\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q514137\">Show Solution<\/button><\/p>\n<div id=\"q514137\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]X=\\left[\\begin{array}{c}4\\\\ 38\\\\ 58\\end{array}\\right][\/latex]<\/p><\/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-egbgcebh-ieFpNMrd9kU\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ieFpNMrd9kU?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-egbgcebh-ieFpNMrd9kU\" 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=12851176&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-egbgcebh-ieFpNMrd9kU&amp;vembed=0&amp;video_id=ieFpNMrd9kU&amp;video_target=tpm-plugin-egbgcebh-ieFpNMrd9kU\" 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\/Using+a+Matrix+Equation+to+Solve+a+System+of+Equations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cUsing a Matrix Equation to Solve a System of Equations\u201d here (opens in new 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