{"id":204,"date":"2017-08-23T09:58:36","date_gmt":"2017-08-23T09:58:36","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=204"},"modified":"2017-09-18T19:21:10","modified_gmt":"2017-09-18T19:21:10","slug":"matrices-operations-and-inverses","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/matrices-operations-and-inverses\/","title":{"rendered":"Matrices\u2014Operations and Inverses"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/proportions-in-measurement\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/measurement-and-linear-algebra\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/matrices-for-systems-of-equations\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Matrices\u2014Operations and Inverses<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will review matrices and their operations. You will also identify the properties of operations on\u00a0matrices and explore inverses.<\/p>\n<h4>Previously Covered:<\/h4>\n<h4>Matrices<\/h4>\n<ul>\n<li>A <em><strong>matrix<\/strong><\/em> is a rectangular array of numbers. The numbers in a matrix are\u00a0referred to as the <em>elements<\/em> of a matrix.<\/li>\n<li>A matrix is made up of rows and columns.The <strong><em>dimensions <\/em><\/strong>of the matrix\u00a0are said to be \u201c<em>m<\/em> by <em>n\u201d <\/em>which is written <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p1_clip_image003.gif\" width=\"51\" height=\"15\" name=\"graphics2\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<li>An\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p1_clip_image006.gif\" width=\"51\" height=\"15\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0matrix has <em>m<\/em> rows and <em>n<\/em> columns. This notation does <em>not<\/em> mean that you multiply <em>m<\/em> by\u00a0<em>n.<\/em><\/li>\n<li>A <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p1_clip_image009.gif\" width=\"44\" height=\"16\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/> matrix is sometimes called <em>a <strong>row\u00a0vector<\/strong><\/em><strong>. <\/strong><\/li>\n<li>A <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p1_clip_image012.gif\" width=\"55\" height=\"16\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>matrix is\u00a0sometimes called a <strong><em>column vector<\/em>.<\/strong><\/li>\n<li>Let <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p2_clip_image003.gif\" width=\"305\" height=\"73\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><em>A<\/em> is a <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p2_clip_image006.gif\" width=\"47\" height=\"16\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/> matrix, while both <em>B<\/em> and <em>C\u00a0<\/em>are <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p2_clip_image009.gif\" width=\"53\" height=\"16\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>matrices.<\/li>\n<li>The number 2 is the entry for <em>a<\/em><sub>11<\/sub>, <em>b<\/em><sub>21<\/sub>, <em>b<\/em><sub>22<\/sub>, and <em>c<\/em><sub>12<\/sub>.<\/li>\n<li>Two matrices <em>A<\/em> and <em>B<\/em> are equal if <em>A<\/em> and <em>B<\/em> have the same dimensions,\u00a0and every entry <em>a<sub>ij<\/sub><\/em> is equal to every entry <em>b<sub>ij<\/sub><\/em>.<\/li>\n<li>The two matrices <em>P<\/em> and <em>Q <\/em>shown below are equal. Notice that they are both <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p2_clip_image012.gif\" width=\"52\" height=\"16\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>matrices and their respective entries are equal to each other.<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p2_clip_image015.gif\" width=\"276\" height=\"87\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ul>\n<section>\n<h3>How do you add and subtract matrices?<\/h3>\n<p>There are no surprises in addition and subtraction of matrices,\u00a0however there is one requirement. The matrices must have the same\u00a0dimensions. Once you establish that the matrices you are working\u00a0with have the same dimensions, you can add or subtract each entry\u00a0in the first matrix with the corresponding entry in the second\u00a0matrix.<\/p>\n<p>The sum and the difference of matrices <em>B<\/em> and <em>C\u00a0<\/em>are shown below.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p2_clip_image018.gif\" width=\"261\" height=\"100\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Let <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p2_clip_image021.gif\" width=\"343\" height=\"97\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>.<br \/>\nWhat is <em>J<\/em> + <em>K<\/em> \u2013 <em>L? <\/em><\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p2_clip_image024.gif\" width=\"80\" height=\"97\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><a name=\"one\"><\/a><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p2_clip_image027.gif\" width=\"84\" height=\"97\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p2_clip_image030.gif\" width=\"84\" height=\"97\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p2_clip_image033.gif\" width=\"80\" height=\"97\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p><span style=\"text-decoration: none;\">The correct choice is B.\u00a0Add each entry <\/span><em><span style=\"text-decoration: none;\">ij\u00a0<\/span><\/em><span style=\"text-decoration: none;\">of <\/span><em><span style=\"text-decoration: none;\">J\u00a0<\/span><\/em><span style=\"text-decoration: none;\">to the respective entry <\/span><em><span style=\"text-decoration: none;\">ij\u00a0<\/span><\/em><span style=\"text-decoration: none;\">of <\/span><em><span style=\"text-decoration: none;\">K,\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and subtract the respective entry <\/span><em><span style=\"text-decoration: none;\">ij\u00a0<\/span><\/em><span style=\"text-decoration: none;\">of <\/span><em><span style=\"text-decoration: none;\">L<\/span><\/em><span style=\"text-decoration: none;\">.<\/span><\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What are the dimensions of the matrix <em>J<\/em> in the example\u00a0above?<\/p>\n<ol>\n<li>\u20132<\/li>\n<li>8<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p2_clip_image036.gif\" width=\"48\" height=\"16\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p2_clip_image039.gif\" width=\"53\" height=\"16\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is D. The dimensions of a matrix are the\u00a0number of rows by the number of columns.<\/p>\n<\/div>\n<\/section>\n<h4>How do you multiply with matrices?<\/h4>\n<p>There are two kinds of multiplication with matrices: <em><strong>scalar\u00a0multiplication<\/strong><\/em> and <em><strong>multiplication of\u00a0matrices<\/strong><\/em>.<\/p>\n<p><strong><em>Scalar multiplication<\/em><\/strong> changes the\u00a0size of every entry by the same factor, called a <em><span style=\"text-decoration: none;\">scalar<\/span><\/em>.\u00a0A matrix of any dimensions may be multiplied by any scalar.\u00a0Consider the scalar <em>r<\/em> = 3 and matrix\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p3_clip_image003.gif\" width=\"93\" height=\"73\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0If we multiply these together we find:<\/p>\n<p align=\"CENTER\"><em>rA<\/em> = <em>r<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p3_clip_image006.gif\" width=\"67\" height=\"73\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/><\/em><br \/>\n= 3<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p3_clip_image009.gif\" width=\"67\" height=\"73\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>=<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p3_clip_image012.gif\" width=\"68\" height=\"73\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p><strong><em>Matrix multiplication <\/em><\/strong>is more\u00a0complicated. Not every pair of matrices can be multiplied. An\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p3_clip_image015.gif\" width=\"56\" height=\"15\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0matrix can be multiplied only by an\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p3_clip_image018.gif\" width=\"47\" height=\"19\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0matrix. This means that the number of columns of the first matrix\u00a0must be the same as the number of rows of the second matrix. If\u00a0the first matrix has dimensions\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p3_clip_image021.gif\" width=\"51\" height=\"15\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and is multiplied by a second matrix of dimensions\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p3_clip_image024.gif\" width=\"47\" height=\"19\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0then the dimensions of the product matrix will be\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p3_clip_image027.gif\" width=\"51\" height=\"19\" name=\"graphics11\" align=\"TOP\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p3_clip_image030.gif\" width=\"331\" height=\"97\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The product <em>WX<\/em> will have dimensions\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p3_clip_image033.gif\" width=\"43\" height=\"16\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>The product <em>XW<\/em> will have dimensions\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p3_clip_image036.gif\" width=\"48\" height=\"16\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>The product <em>WY<\/em> will have dimensions\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/s5_p3_html_61ca6d1e.gif\" width=\"41\" height=\"17\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><span style=\"text-decoration: none;\">The product <em>YW\u00a0<\/em>cannot be determined because the number of columns of <\/span><em><span style=\"text-decoration: none;\">Y\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is not equal to the number of rows of <\/span><em>W<\/em>. The product\u00a0<em>YW<\/em> is not defined.<\/p>\n<p>&nbsp;<\/p>\n<p align=\"CENTER\"><strong>Multiplying Matrices<\/strong><\/p>\n<p align=\"center\"><span style=\"text-decoration: none;\">To find the product of <\/span><em><span style=\"text-decoration: none;\">AB<\/span><\/em><span style=\"text-decoration: none;\">: Each entry in the product matrix is a sum of the products of each entry in row <\/span><em><span style=\"text-decoration: none;\">i <\/span><\/em><span style=\"text-decoration: none;\">of <\/span><em><span style=\"text-decoration: none;\">A<\/span><\/em><span style=\"text-decoration: none;\"> multiplied by the corresponding entry of column <\/span><em><span style=\"text-decoration: none;\">j<\/span><\/em><span style=\"text-decoration: none;\"> of B. More precisely: <\/span><\/p>\n<p align=\"CENTER\">If <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p3_clip_image042.gif\" width=\"175\" height=\"49\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>,<\/p>\n<p align=\"CENTER\"><span style=\"text-decoration: none;\">then <\/span><em><span style=\"text-decoration: none;\">AB<\/span><\/em> = <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p3_clip_image045.gif\" width=\"255\" height=\"49\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the product <em>GH<\/em>, where<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image003.gif\" width=\"209\" height=\"49\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image006.gif\" width=\"69\" height=\"49\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image009.gif\" width=\"61\" height=\"49\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image012.gif\" width=\"84\" height=\"49\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image015.gif\" width=\"84\" height=\"49\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is D.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image018.gif\" width=\"673\" height=\"49\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is <em>ST <\/em>+ <em>P<\/em>, where<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image021.gif\" width=\"308\" height=\"49\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/>?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image024.gif\" width=\"75\" height=\"49\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image027.gif\" width=\"59\" height=\"49\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image030.gif\" width=\"68\" height=\"49\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image033.gif\" width=\"76\" height=\"49\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is B.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image036.gif\" width=\"505\" height=\"49\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is <em><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image039.gif\" width=\"21\" height=\"20\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/><\/em>,\u00a0where <em>A<\/em> =\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image042.gif\" width=\"60\" height=\"49\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image045.gif\" width=\"60\" height=\"49\" name=\"graphics17\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image048.gif\" width=\"60\" height=\"49\" name=\"graphics18\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image051.gif\" width=\"60\" height=\"49\" name=\"graphics19\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image054.gif\" width=\"60\" height=\"49\" name=\"graphics20\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is A.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image057.gif\" width=\"420\" height=\"49\" name=\"graphics21\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is 2<em>BA<\/em> + <em>C<\/em><sup>2<\/sup>, where <em>A<\/em> =<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image060.gif\" width=\"280\" height=\"49\" name=\"graphics22\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image063.gif\" width=\"60\" height=\"49\" name=\"graphics23\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image066.gif\" width=\"60\" height=\"49\" name=\"graphics24\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image069.gif\" width=\"51\" height=\"49\" name=\"graphics25\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image072.gif\" width=\"67\" height=\"49\" name=\"graphics26\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is B.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image075.gif\" width=\"528\" height=\"98\" name=\"graphics27\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p><span style=\"text-decoration: none;\">What is 3<\/span><em><span style=\"text-decoration: none;\">GH<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0where <\/span><em><span style=\"text-decoration: none;\">G\u00a0<\/span><\/em>=\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image078.gif\" width=\"140\" height=\"73\" name=\"graphics28\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image081.gif\" width=\"45\" height=\"73\" name=\"graphics29\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image084.gif\" width=\"44\" height=\"73\" name=\"graphics30\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image087.gif\" width=\"44\" height=\"73\" name=\"graphics31\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image090.gif\" width=\"75\" height=\"97\" name=\"graphics32\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is B.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p4_clip_image093.gif\" width=\"228\" height=\"73\" name=\"graphics33\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<h3><strong>Identity Matrix <\/strong><\/h3>\n<p>When adding, an <strong><em>identity matrix<\/em><\/strong> is a matrix whose entries are\u00a0all zeroes. This identity matrix is denoted <em>O<\/em>. Some\u00a0examples are as follows:<\/p>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p5_clip_image003.gif\" width=\"203\" height=\"73\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/p>\n<p>When multiplying, an identity matrix is a matrix whose entries\u00a0are all zeroes, except for the entries that lie on the main\u00a0diagonal (top left to bottom right). The entries on the main\u00a0diagonal are all ones. The identity matrix for multiplication is\u00a0denoted <em>I<\/em>. Some examples are as follows.<\/p>\n<p align=\"CENTER\"><em><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p5_clip_image006.gif\" width=\"253\" height=\"97\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/em><\/p>\n<h4>How do you divide matrices?<\/h4>\n<p>Instead of division, matrix algebra focuses on using the\u00a0multiplicative inverse of a matrix. The inverse of <em>A<\/em>,\u00a0denoted by <em>A<\/em><sup>\u20131<\/sup>, is the matrix that makes the\u00a0expression below true.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p5_clip_image009.gif\" width=\"109\" height=\"20\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<table border=\"2\" width=\"75%\" cellpadding=\"10\" align=\"center\">\n<tbody>\n<tr>\n<td>\n<p class=\"notebox_text\" align=\"CENTER\"><strong>Important Tidbit<\/strong><\/p>\n<p class=\"notebox_text\" align=\"center\">In order to have an inverse, a matrix must be square. Not all square<br \/>\nmatrices have an inverse.<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>For example, let <em>T<\/em> =\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p5_clip_image012.gif\" width=\"68\" height=\"49\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0To find <em>T<\/em><sup>\u20131<\/sup>, solve the following:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p5_clip_image015.gif\" width=\"180\" height=\"49\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>This means:<\/p>\n<p align=\"CENTER\"><span style=\"text-decoration: none;\">\u20132<\/span><em><span style=\"text-decoration: none;\">a\u00a0<\/span><\/em><span style=\"text-decoration: none;\">+ 3<\/span><em><span style=\"text-decoration: none;\">c\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= 1\u00a0<\/span><span style=\"text-decoration: none;\">\u20132<\/span><em><span style=\"text-decoration: none;\">b\u00a0<\/span><\/em><span style=\"text-decoration: none;\">+ 3<\/span><em><span style=\"text-decoration: none;\">d\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= 0\u00a0<\/span><em><span style=\"text-decoration: none;\">a<\/span><\/em><span style=\"text-decoration: none;\"> \u2013 2<\/span><em><span style=\"text-decoration: none;\">c\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= 0\u00a0<\/span><em><span style=\"text-decoration: none;\">b<\/span><\/em><span style=\"text-decoration: none;\"> \u2013 2<\/span><em><span style=\"text-decoration: none;\">d <\/span><\/em><span style=\"text-decoration: none;\">=\u00a01,<\/span><\/p>\n<p align=\"CENTER\"><em><span style=\"text-decoration: none;\">a<\/span><\/em><span style=\"text-decoration: none;\"> = \u20132, <\/span><em><span style=\"text-decoration: none;\">b<\/span><\/em><span style=\"text-decoration: none;\"> = \u20133, <\/span><em><span style=\"text-decoration: none;\">c<\/span><\/em><span style=\"text-decoration: none;\"> = \u20131, and <\/span><em><span style=\"text-decoration: none;\">d<\/span><\/em><span style=\"text-decoration: none;\"> = \u20132. <\/span><\/p>\n<p align=\"CENTER\"><span style=\"text-decoration: none;\">Thus<\/span>,<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p5_clip_image018.gif\" width=\"107\" height=\"49\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>In fact, the inverse of a square matrix can be described by<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p5_clip_image021.gif\" width=\"159\" height=\"49\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0where <em>ad<\/em> \u2013 <em>bc <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s5_p5_clip_image024.gif\" width=\"15\" height=\"16\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/>0.<\/em><\/p>\n<p>For the square matrix above, the value\u00a0<em><span style=\"text-decoration: none;\">ad \u2013 bc\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is called the <\/span><em><span style=\"text-decoration: none;\">determinant<\/span><\/em><span style=\"text-decoration: none;\">.\u00a0A matrix is invertible only if its determinant is non-zero.\u00a0Determinants for larger matric<\/span>es are discussed in another\u00a0unit.<\/p>\n<table width=\"75%\">\n<thead>\n<tr valign=\"ABSMIDDLE\">\n<th colspan=\"3\">\n<p align=\"CENTER\">Properties of Matrix Algebra<\/p>\n<\/th>\n<\/tr>\n<tr valign=\"TOP\">\n<th><\/th>\n<th class=\"lesson_subhead\" width=\"237\">\n<p align=\"center\">Addition<\/p>\n<\/th>\n<th class=\"lesson_subhead\">\n<p align=\"center\">Multiplication<\/p>\n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"TOP\">\n<td><strong>Associative Property <\/strong><\/td>\n<td width=\"237\">Yes<\/td>\n<td colspan=\"2\">Yes<\/td>\n<\/tr>\n<tr valign=\"TOP\">\n<td><strong>Commutative Property <\/strong><\/td>\n<td width=\"237\">Yes<\/td>\n<td>No<\/td>\n<\/tr>\n<tr valign=\"TOP\">\n<td><strong>Identity <\/strong><\/td>\n<td width=\"237\">A matrix in which all entries are zero. This matrix is denoted O.<\/td>\n<td>A square matrix in which all entries are zero except the entries on the diagonal from<br \/>\n<em>a<\/em><sub>11<\/sub> to <em>a<sub>nn<\/sub> ,<\/em> which has entries of 1. This matrix is denoted <em>I<\/em>.<\/td>\n<\/tr>\n<tr valign=\"TOP\">\n<td><strong>Inverses <\/strong><\/td>\n<td width=\"237\">The inverse of <em>A<\/em> is the scalar product of \u20131(<em>A<\/em>).<\/td>\n<td>Only some matrices have inverses.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Review of New Vocabulary and Concepts<\/h3>\n<ul>\n<li>A <em><strong>matrix\u00a0<\/strong><\/em>is a rectangular array of numbers.\u00a0Matrices can be added,\u00a0subtracted, and multiplied.<\/li>\n<li>The associative property exists\u00a0in both addition and multiplication of matrices.<\/li>\n<li>The commutative property exists\u00a0only in matrix addition.<\/li>\n<li>There are identity matrices in\u00a0both addition and multiplication.<\/li>\n<li>Every matrix has an additive\u00a0inverse.<\/li>\n<li>Only square matrices have\u00a0multiplicative inverses.<\/li>\n<li>Not all square matrices have multiplicative inverses.<\/li>\n<\/ul>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/proportions-in-measurement\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/measurement-and-linear-algebra\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/matrices-for-systems-of-equations\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Matrices\u2014Operations and Inverses Objective In this lesson, you will review matrices and their operations. You will also identify the properties of operations on\u00a0matrices and explore inverses. Previously Covered: Matrices A matrix is a rectangular array of numbers. The numbers in a matrix are\u00a0referred to as the elements of a [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-204","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/204","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=204"}],"version-history":[{"count":11,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/204\/revisions"}],"predecessor-version":[{"id":768,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/204\/revisions\/768"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=204"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}