{"id":136,"date":"2025-02-13T22:44:11","date_gmt":"2025-02-13T22:44:11","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/matrices-and-matrix-operations\/"},"modified":"2026-03-23T17:32:17","modified_gmt":"2026-03-23T17:32:17","slug":"matrices-and-matrix-operations","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/matrices-and-matrix-operations\/","title":{"raw":"Matrices and Matrix Operations: Learn It 1","rendered":"Matrices and Matrix Operations: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><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\nTwo club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season. The table\u00a0shows the needs of both teams.\r\n<table summary=\"..\">\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th style=\"text-align: center;\">Wildcats<\/th>\r\n<th style=\"text-align: center;\">Mud Cats<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td><strong>Goals<\/strong><\/td>\r\n<td style=\"text-align: center;\">[latex]6[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Balls<\/strong><\/td>\r\n<td style=\"text-align: center;\">[latex]30[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]24[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Jerseys<\/strong><\/td>\r\n<td style=\"text-align: center;\">[latex]14[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]20[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nA goal costs [latex]$300[\/latex]; a ball costs [latex]$10[\/latex]; and a jersey costs [latex]$30[\/latex]. How can we find the total cost for the equipment needed for each team?\r\n\r\nTo solve a problem like this, we can use a <strong>matrix<\/strong>, which is a rectangular array of numbers. A <strong>row<\/strong> in a matrix is a set of numbers that are aligned horizontally. A <strong>column<\/strong> in a matrix is a set of numbers that are aligned vertically. Each number is an <strong>entry<\/strong>, sometimes called an element, of the matrix. Matrices (plural) are enclosed in [ ] or ( ) and are usually named with capital letters.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">For example, three matrices named [latex]A,B,\\text{}[\/latex] and [latex]C[\/latex] are shown below.\r\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}1&amp; 2\\\\ 3&amp; 4\\end{array}\\right],B=\\left[\\begin{array}{ccc}\\hfill 1&amp; \\hfill 2&amp; \\hfill 7\\\\ \\hfill 0&amp; \\hfill -5&amp; \\hfill 6\\\\ \\hfill 7&amp; \\hfill 8&amp; \\hfill 2\\end{array}\\right],C=\\left[\\begin{array}{c}\\hfill -1\\\\ \\hfill 0\\\\ \\hfill 3\\end{array}\\begin{array}{c}3\\\\ 2\\\\ 1\\end{array}\\right][\/latex]<\/p>\r\n\r\n<\/section>A matrix is often referred to by its size or dimensions: [latex]\\text{ }m\\text{ }\\times \\text{ }n\\text{ }[\/latex] indicating [latex]m[\/latex] rows and [latex]n[\/latex] columns. Matrix entries are defined first by row and then by column.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">For example, to locate the entry in matrix [latex]A[\/latex] identified as [latex]{a}_{ij},\\text{}[\/latex] we look for the entry in row [latex]i,\\text{}[\/latex] column [latex]j[\/latex]. In matrix [latex]A[\/latex] shown below, the entry in row 2, column 3 is [latex]{a}_{23}[\/latex].\r\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}{a}_{11}&amp; {a}_{12}&amp; {a}_{13}\\\\ {a}_{21}&amp; {a}_{22}&amp; {a}_{23}\\\\ {a}_{31}&amp; {a}_{32}&amp; {a}_{33}\\end{array}\\right][\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>matrix<\/h3>\r\nA <strong>matrix<\/strong> is a rectangular array of numbers arranged in rows and columns.\r\n<div class=\"page\" title=\"Page 878\">\r\n<div class=\"layoutArea\">\r\n<div class=\"column\">\r\n\r\n&nbsp;\r\n\r\nA matrix with [latex]m[\/latex] rows and [latex]n[\/latex] columns has dimension [latex]m \\times n[\/latex].\r\n<div class=\"page\" title=\"Page 878\">\r\n<div class=\"layoutArea\">\r\n<div class=\"column\">\r\n\r\nEach number in the matrix is called an <strong>element<\/strong> or <strong>entry<\/strong> in the matrix.\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<img class=\"aligncenter size-full wp-image-2442\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/29205819\/Screenshot-2024-07-29-at-1.58.15%E2%80%AFPM.png\" alt=\"\" width=\"800\" height=\"274\" \/>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<h3>Type of Matrices<\/h3>\r\n<ul>\r\n \t<li>A <strong>square matrix<\/strong> is a matrix with dimensions [latex]\\text{ }n\\text{ }\\times \\text{ }n,\\text{}[\/latex] meaning that it has the same number of rows as columns:\r\n<center>[latex]A=\\left[\\begin{array}{ccc}{a}_{11}&amp; {a}_{12}&amp; {a}_{13}\\\\ {a}_{21}&amp; {a}_{22}&amp; {a}_{23}\\\\ {a}_{31}&amp; {a}_{32}&amp; {a}_{33}\\end{array}\\right][\/latex]<\/center><\/li>\r\n \t<li>A <strong>row matrix<\/strong> is a matrix consisting of one row with dimensions [latex]1\\text{ }\\times \\text{ }n[\/latex]:\r\n<center>[latex]\\left[\\begin{array}{ccc}{a}_{11}&amp; {a}_{12}&amp; {a}_{13}\\end{array}\\right][\/latex]<\/center><\/li>\r\n \t<li>A <strong>column matrix<\/strong> is a matrix consisting of one column with dimensions [latex]m\\text{ }\\times \\text{ }1[\/latex]:\r\n<center>[latex]\\left[\\begin{array}{c}{a}_{11}\\\\ {a}_{21}\\\\ {a}_{31}\\end{array}\\right][\/latex]<\/center><\/li>\r\n<\/ul>\r\n<section class=\"textbox example\" aria-label=\"Example\">Given matrix [latex]A:[\/latex]\r\n<ol>\r\n \t<li>What are the dimensions of matrix [latex]A?[\/latex]<\/li>\r\n \t<li>What are the entries at [latex]{a}_{31}[\/latex] and [latex]{a}_{22}?[\/latex]<\/li>\r\n<\/ol>\r\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{rrrr}\\hfill 2&amp; \\hfill &amp; \\hfill 1&amp; \\hfill 0\\\\ \\hfill 2&amp; \\hfill &amp; \\hfill 4&amp; \\hfill 7\\\\ \\hfill 3&amp; \\hfill &amp; \\hfill 1&amp; \\hfill -2\\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"832047\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"832047\"]\r\n<ol>\r\n \t<li>The dimensions are [latex]\\text{ }3\\text{ }\\times \\text{ }3\\text{ }[\/latex] because there are three rows and three columns.<\/li>\r\n \t<li>Entry [latex]{a}_{31}[\/latex] is the number at row 3, column 1 which is [latex]3[\/latex]. The entry [latex]{a}_{22}[\/latex] is the number at row 2, column 2 which is [latex]4[\/latex]. <em>Remember, the row comes first, then the column.<\/em><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321695[\/ohm_question]<\/section><\/div>\r\n<dl id=\"fs-id1165135199312\" class=\"definition\">\r\n \t<dd id=\"fs-id1165135199316\"><\/dd>\r\n<\/dl>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<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<p>Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season. The table\u00a0shows the needs of both teams.<\/p>\n<table summary=\"..\">\n<thead>\n<tr>\n<th><\/th>\n<th style=\"text-align: center;\">Wildcats<\/th>\n<th style=\"text-align: center;\">Mud Cats<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><strong>Goals<\/strong><\/td>\n<td style=\"text-align: center;\">[latex]6[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]10[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Balls<\/strong><\/td>\n<td style=\"text-align: center;\">[latex]30[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]24[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Jerseys<\/strong><\/td>\n<td style=\"text-align: center;\">[latex]14[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]20[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>A goal costs [latex]$300[\/latex]; a ball costs [latex]$10[\/latex]; and a jersey costs [latex]$30[\/latex]. How can we find the total cost for the equipment needed for each team?<\/p>\n<p>To solve a problem like this, we can use a <strong>matrix<\/strong>, which is a rectangular array of numbers. A <strong>row<\/strong> in a matrix is a set of numbers that are aligned horizontally. A <strong>column<\/strong> in a matrix is a set of numbers that are aligned vertically. Each number is an <strong>entry<\/strong>, sometimes called an element, of the matrix. Matrices (plural) are enclosed in [ ] or ( ) and are usually named with capital letters.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">For example, three matrices named [latex]A,B,\\text{}[\/latex] and [latex]C[\/latex] are shown below.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}1& 2\\\\ 3& 4\\end{array}\\right],B=\\left[\\begin{array}{ccc}\\hfill 1& \\hfill 2& \\hfill 7\\\\ \\hfill 0& \\hfill -5& \\hfill 6\\\\ \\hfill 7& \\hfill 8& \\hfill 2\\end{array}\\right],C=\\left[\\begin{array}{c}\\hfill -1\\\\ \\hfill 0\\\\ \\hfill 3\\end{array}\\begin{array}{c}3\\\\ 2\\\\ 1\\end{array}\\right][\/latex]<\/p>\n<\/section>\n<p>A matrix is often referred to by its size or dimensions: [latex]\\text{ }m\\text{ }\\times \\text{ }n\\text{ }[\/latex] indicating [latex]m[\/latex] rows and [latex]n[\/latex] columns. Matrix entries are defined first by row and then by column.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">For example, to locate the entry in matrix [latex]A[\/latex] identified as [latex]{a}_{ij},\\text{}[\/latex] we look for the entry in row [latex]i,\\text{}[\/latex] column [latex]j[\/latex]. In matrix [latex]A[\/latex] shown below, the entry in row 2, column 3 is [latex]{a}_{23}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\\\ {a}_{21}& {a}_{22}& {a}_{23}\\\\ {a}_{31}& {a}_{32}& {a}_{33}\\end{array}\\right][\/latex]<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>matrix<\/h3>\n<p>A <strong>matrix<\/strong> is a rectangular array of numbers arranged in rows and columns.<\/p>\n<div class=\"page\" title=\"Page 878\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p>&nbsp;<\/p>\n<p>A matrix with [latex]m[\/latex] rows and [latex]n[\/latex] columns has dimension [latex]m \\times n[\/latex].<\/p>\n<div class=\"page\" title=\"Page 878\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p>Each number in the matrix is called an <strong>element<\/strong> or <strong>entry<\/strong> in the matrix.<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2442\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/29205819\/Screenshot-2024-07-29-at-1.58.15%E2%80%AFPM.png\" alt=\"\" width=\"800\" height=\"274\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h3>Type of Matrices<\/h3>\n<ul>\n<li>A <strong>square matrix<\/strong> is a matrix with dimensions [latex]\\text{ }n\\text{ }\\times \\text{ }n,\\text{}[\/latex] meaning that it has the same number of rows as columns:\n<div style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\\\ {a}_{21}& {a}_{22}& {a}_{23}\\\\ {a}_{31}& {a}_{32}& {a}_{33}\\end{array}\\right][\/latex]<\/div>\n<\/li>\n<li>A <strong>row matrix<\/strong> is a matrix consisting of one row with dimensions [latex]1\\text{ }\\times \\text{ }n[\/latex]:\n<div style=\"text-align: center;\">[latex]\\left[\\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\end{array}\\right][\/latex]<\/div>\n<\/li>\n<li>A <strong>column matrix<\/strong> is a matrix consisting of one column with dimensions [latex]m\\text{ }\\times \\text{ }1[\/latex]:\n<div style=\"text-align: center;\">[latex]\\left[\\begin{array}{c}{a}_{11}\\\\ {a}_{21}\\\\ {a}_{31}\\end{array}\\right][\/latex]<\/div>\n<\/li>\n<\/ul>\n<section class=\"textbox example\" aria-label=\"Example\">Given matrix [latex]A:[\/latex]<\/p>\n<ol>\n<li>What are the dimensions of matrix [latex]A?[\/latex]<\/li>\n<li>What are the entries at [latex]{a}_{31}[\/latex] and [latex]{a}_{22}?[\/latex]<\/li>\n<\/ol>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{rrrr}\\hfill 2& \\hfill & \\hfill 1& \\hfill 0\\\\ \\hfill 2& \\hfill & \\hfill 4& \\hfill 7\\\\ \\hfill 3& \\hfill & \\hfill 1& \\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=\"q832047\">Show Solution<\/button><\/p>\n<div id=\"q832047\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The dimensions are [latex]\\text{ }3\\text{ }\\times \\text{ }3\\text{ }[\/latex] because there are three rows and three columns.<\/li>\n<li>Entry [latex]{a}_{31}[\/latex] is the number at row 3, column 1 which is [latex]3[\/latex]. The entry [latex]{a}_{22}[\/latex] is the number at row 2, column 2 which is [latex]4[\/latex]. <em>Remember, the row comes first, then the column.<\/em><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321695\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321695&theme=lumen&iframe_resize_id=ohm321695&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n<dl id=\"fs-id1165135199312\" class=\"definition\">\n<dd id=\"fs-id1165135199316\"><\/dd>\n<\/dl>\n","protected":false},"author":6,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":514,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/136"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/136\/revisions"}],"predecessor-version":[{"id":5964,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/136\/revisions\/5964"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/514"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/136\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=136"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=136"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=136"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=136"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}