{"id":61701,"date":"2020-09-14T19:20:22","date_gmt":"2020-09-14T19:20:22","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=61701"},"modified":"2026-03-26T13:00:23","modified_gmt":"2026-03-26T18:00:23","slug":"matrices-the-basics","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/matrices-the-basics\/","title":{"rendered":"Matrices: The Basics"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_Us3II24rYFI\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_Us3II24rYFI\" data-source-videoID=\"Us3II24rYFI\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Matrices: The Basics Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Matrices: The Basics\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_Us3II24rYFI:hover {cursor:pointer;} img#videoThumbnailImage_Us3II24rYFI {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/1682-matrices-the-basics-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_Us3II24rYFI\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_Us3II24rYFI\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Matrices: The Basics\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_Us3II24rYFI\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_Us3II24rYFI\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_Us3II24rYFI\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction DBY_Function() {\n  var x = document.getElementById(\"DBY\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"DBY_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"DBY\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#What_is_a_Matrix\" class=\"smooth-scroll\">What is a Matrix?<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Square_Matrix_Examples\" class=\"smooth-scroll\">Square Matrix Examples<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Adding_and_Subtracting_Matrices\" class=\"smooth-scroll\">Adding and Subtracting Matrices<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Multiplying_Matrices\" class=\"smooth-scroll\">Multiplying Matrices<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Matrices_%E2%80%93_Practice_Questions\" class=\"smooth-scroll\">Matrices \u2013 Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hello, and welcome to this video about the basics of matrices! <\/p>\n<p>In this video, we will cover:<\/p>\n<ul>\n<li>What matrices are<\/li>\n<li>Some components of matrices<\/li>\n<li>Some types of matrices<\/li>\n<li>Matrix operations<\/li>\n<\/ul>\n<h2><span id=\"What_is_a_Matrix\" class=\"m-toc-anchor\"><\/span>What is a Matrix?<\/h2>\n<p>\nA <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/matrices-transposition-determinants-and-augmentation\/\">matrix<\/a> is commonly defined as a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices can be used in several different algebraic and geometric situations.<\/p>\n<p>Here\u2019s the general form of a matrix:<\/p>\n<p>\\begin{bmatrix}<br \/>\na_{1,1} &#038; a_{1,2} &#038; a_{1,3} &#038; \\cdots \\\\<br \/>\na_{2,1} &#038; a_{2,2} &#038; a_{2,3} &#038; \\cdots \\\\<br \/>\na_{3,1} &#038; a_{3,2} &#038; a_{3,3} &#038; \\cdots \\\\<br \/>\n\\vdots  &#038; \\vdots  &#038; \\vdots  &#038; \\ddots<br \/>\n\\end{bmatrix}<\/p>\n<p>As you can see, it is made of an undetermined number of rows and columns.<\/p>\n<p>An example of a matrix might look like this:<\/p>\n<p>\\[<br \/>\nA =<br \/>\n\\left[<br \/>\n\\begin{array}{rr} <br \/>\n  -1 &#038; 5  \\\\<br \/>\n   6 &#038; -3 \\\\<br \/>\n   7 &#038; 2  <br \/>\n\\end{array}<br \/>\n\\right]<br \/>\n\\]<\/p>\n<p>Matrices are usually written in square brackets and they are commonly classified by their <strong>dimension<\/strong>. Matrix A is a 3\u00d72 matrix because it consists of 3 rows and 2 columns.<\/p>\n<p>Each of the numbers in matrix A, notated as [A], is called an <strong>element<\/strong> or an entry. Each element can be classified according to its position within [A]. \\(A_{m,n}\\) designates the element at row \\(m\\) column \\(n\\). For example, \\(A_{2,1}=6\\) because 6 is the element located at row 2 column 1 of the matrix.<\/p>\n<p>Some matrices consist of only one row or column. These are known as vectors.<\/p>\n<div class=\"matrixcolumn-container\">\n<div class=\"matrixleftcolumn\">\n\\[<br \/>\nB =<br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 48 \\\\<br \/>\n\\end{bmatrix}<br \/>\n\\]\n<\/div>\n<div class=\"matrixrightcolumn\">\n\\[<br \/>\nC =<br \/>\n\\begin{bmatrix}<br \/>\n2 \\\\<br \/>\n5 \\\\<br \/>\n11<br \/>\n\\end{bmatrix}<br \/>\n\\]\n<\/div>\n<\/div>\n<p>\n&nbsp;<br \/>\nMatrix B might also be classified as a row matrix, and Matrix C might also be classified as a column matrix. Notice that Matrix B is a 1\u00d73 matrix, while Matrix C is a 3\u00d71 matrix.<\/p>\n<p>The elements where \\(m=n\\) constitute the main diagonal of a matrix. In this matrix here, the elements \\(A_{1,1}\\) and \\(A_{2,2}\\) are on the main diagonal (the entries negative 1 and negative 3 constitute the main diagonal):<\/p>\n<p>\\[<br \/>\nA =<br \/>\n\\left[<br \/>\n\\begin{array}{rr} <br \/>\n  -1 &#038; 5  \\\\<br \/>\n   6 &#038; -3 \\\\<br \/>\n   7 &#038; 2  <br \/>\n\\end{array}<br \/>\n\\right]<br \/>\n\\]<\/p>\n<h2 style=\"margin-top: 2em;\"><span id=\"Square_Matrix_Examples\" class=\"m-toc-anchor\"><\/span>Square Matrix Examples<\/h2>\n<p>\nIn matrices with unequal numbers of rows and columns, the main diagonal isn\u2019t nearly as useful as it is with square matrices (matrices with an equal number of rows and columns). Here are some examples of square matrices.<\/p>\n<p>Matrix D might be called a <strong>lower triangular matrix<\/strong>, because all the entries above the main diagonal are 0:<\/p>\n<p>\\[<br \/>\nD =<br \/>\n\\left[<br \/>\n\\begin{array}{r}<br \/>\n-2 &#038; 0 &#038; 0 \\\\<br \/>\n-6  &#038; 7 &#038; 0 \\\\<br \/>\n-9  &#038; 12 &#038; 14<br \/>\n\\end{array}<br \/>\n\\right]<br \/>\n\\]<\/p>\n<p>Matrix E might be called an <strong>upper triangular matrix<\/strong> because all the entries below the main diagonal are 0:<\/p>\n<p>\\[<br \/>\nE =<br \/>\n\\begin{bmatrix}<br \/>\n0 &#038; 99 &#038; \\phantom{-}2 \\\\<br \/>\n0  &#038; 1 &#038; -15 \\\\<br \/>\n0  &#038; 0 &#038; \\phantom{-}4<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>Note that triangular matrices can have entries that are 0 on the main diagonal, like [E] does.<\/p>\n<p>Matrix F has the qualities of both [D] and [E]:<\/p>\n<p>\\[<br \/>\nF =<br \/>\n\\left[<br \/>\n\\begin{array}{rrcc}<br \/>\n    5  &#038;  0  &#038;  0  &#038;  0  \\\\<br \/>\n    0  &#038; -3  &#038;  0  &#038;  0  \\\\<br \/>\n    0  &#038;  0  &#038;  4  &#038;  0  \\\\<br \/>\n    0  &#038;  0  &#038;  0  &#038; 12<br \/>\n\\end{array}<br \/>\n\\right]<br \/>\n\\]<\/p>\n<p>The entries are 0 everywhere but the main diagonal. Instead of calling this matrix triangular, we simply call it diagonal.<\/p>\n<p>Diagonal matrices whose entries on the main diagonal are all 1s are called <strong>identity matrices<\/strong>. Matrix G is known as the 2\u00d72 identity matrix, notated by \\(I_{2}\\):<\/p>\n<p>\\[<br \/>\nH =<br \/>\n\\begin{bmatrix}<br \/>\n0 &#038; 0 \\\\<br \/>\n0  &#038; 0<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>Sometimes, matrices like [H] have entries that are all 0s. [H] is known as the 2\u00d72 <strong>zero matrix<\/strong> or null matrix, which is notated as \\(0_{2&#215;2}\\).<\/p>\n<p>In order for matrices to be equal, they must contain the same elements in the same positions.<\/p>\n<p>In this case, [I]=[K], but [I] \u2260[J] and [J] \u2260 [K]:<\/p>\n<div class=\"matrixcolumn-container\">\n<div class=\"matrixleftcolumn\">\n\\[<br \/>\nI =<br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 2 \\\\<br \/>\n3  &#038; 4<br \/>\n\\end{bmatrix}<br \/>\n\\]\n<\/div>\n<div class=\"matrixleftcolumn\">\n\\[<br \/>\nJ =<br \/>\n\\begin{bmatrix}<br \/>\n2 &#038; 3 \\\\<br \/>\n1  &#038; 4<br \/>\n\\end{bmatrix}<br \/>\n\\]\n<\/div>\n<div class=\"matrixrightcolumn\">\n\\[<br \/>\nK =<br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 2 \\\\<br \/>\n3  &#038; 4<br \/>\n\\end{bmatrix}<br \/>\n\\]\n<\/div>\n<\/div>\n<p>\n&nbsp;<\/p>\n<h2><span id=\"Adding_and_Subtracting_Matrices\" class=\"m-toc-anchor\"><\/span>Adding and Subtracting Matrices<\/h2>\n<p>\nMatrices can be added and subtracted if they have the same dimension. To add or subtract matrices, simply add or subtract the corresponding elements.<\/p>\n<h3><span id=\"Addition\" class=\"m-toc-anchor\"><\/span>Addition<\/h3>\n<div class=\"matrixcolumn-container\">\n<div class=\"matrixleftcolumn-long\">\n\\[<br \/>\nD =<br \/>\n\\left[<br \/>\n\\begin{array}{r}<br \/>\n-2 &#038; 0 &#038; 0 \\\\<br \/>\n-6  &#038; 7 &#038; 0 \\\\<br \/>\n-9  &#038; 12 &#038; 14<br \/>\n\\end{array}<br \/>\n \\right]<br \/>\n\\]\n<\/div>\n<div class=\"matrixrightcolumn-long\">\n\\[<br \/>\nE =<br \/>\n\\begin{bmatrix}<br \/>\n0 &#038; 99 &#038; \\phantom{-}2 \\\\<br \/>\n0  &#038; 1 &#038; -15 \\\\<br \/>\n0  &#038; 0 &#038; \\phantom{-}4<br \/>\n\\end{bmatrix}<br \/>\n\\]\n<\/div>\n<\/div>\n<p>\n&nbsp;<br \/>\n[D] is a 3&#215;3 matrix, as is [E], so they can be added or subtracted. To add Matrix D plus Matrix E, you&#8217;re just going to add the corresponding components: <\/p>\n<div class=\"longmatrix-container\">\n\\[<br \/>\nD+E =<br \/>\n\\begin{bmatrix}<br \/>\n\\phantom{-}2+0 &#038; 0+99 &#038; 0+2 \\\\<br \/>\n\\phantom{-}6+0 &#038; 7+1 &#038; 0+(-15) \\\\<br \/>\n-9 +0 &#038; 12+0 &#038; 14+4<br \/>\n\\end{bmatrix}<br \/>\n\\]\n<\/div>\n<p>\nFrom there, you just simplify!<\/p>\n<p>\\[<br \/>\nD+E =<br \/>\n\\begin{bmatrix}<br \/>\n \\phantom{-}2 &#038; 99 &#038;  \\phantom{-}2 \\\\<br \/>\n \\phantom{-}6 &#038; 8 &#038; -15 \\\\<br \/>\n-9 &#038; 12 &#038;  \\phantom{-}18<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<h3><span id=\"Subtraction\" class=\"m-toc-anchor\"><\/span>Subtraction<\/h3>\n<p>\nFor subtraction, you&#8217;ll do the same steps but with the different operation. This time let&#8217;s do [E] minus [D].<\/p>\n<div class=\"longmatrix-container\">\n\\[<br \/>\nE-D =<br \/>\n\\begin{bmatrix}<br \/>\n0-2 &#038; 99-0 &#038; 2-0 \\\\<br \/>\n0-6 &#038; 1-7 &#038; -15-0 \\\\<br \/>\n0-(-9) &#038; 0-12 &#038; 4-14<br \/>\n\\end{bmatrix}<br \/>\n\\]\n<\/div>\n<p>\nFrom here, you just simplify!<\/p>\n<p>\\[<br \/>\nE-D =<br \/>\n\\begin{bmatrix}<br \/>\n-2 &#038;  \\phantom{-}99 &#038;  \\phantom{-}2 \\\\<br \/>\n-6 &#038; -6 &#038; -15 \\\\<br \/>\n \\phantom{-}9 &#038; -12 &#038; -10<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>Remember, when adding or subtracting matrices they must be the same size. Then, all you do is add or subtract the matching components for each matrix.<\/p>\n<h2><span id=\"Multiplying_Matrices\" class=\"m-toc-anchor\"><\/span>Multiplying Matrices<\/h2>\n<h3><span id=\"Multiplying_by_a_Scalar\" class=\"m-toc-anchor\"><\/span>Multiplying by a Scalar<\/h3>\n<p>\nMatrices can be multiplied by a scalar.<\/p>\n<p>To multiply a matrix by a scalar, simply multiply each element of the matrix by the scalar. <\/p>\n<p>\\[<br \/>\n3<br \/>\n\\left[<br \/>\n\\begin{array}{r} <br \/>\n2 &#038; 0 &#038; 0 \\\\<br \/>\n6 &#038; 7 &#038; 0 \\\\<br \/>\n-9 &#038; 12 &#038; 14<br \/>\n\\end{array}<br \/>\n\\right]<br \/>\n\\]<\/p>\n<p>So what we&#8217;re going to do is multiply 3 by each of these different elements.<\/p>\n<p>\\begin{bmatrix}<br \/>\n3\\cdot 2 &#038; 3\\cdot 0 &#038; 3\\cdot 0 \\\\<br \/>\n3\\cdot 6 &#038; 3\\cdot 7 &#038; 3\\cdot 0 \\\\<br \/>\n3\\cdot (-9) &#038; 3\\cdot 12 &#038; 3\\cdot 14<br \/>\n\\end{bmatrix}<\/p>\n<p>Then, simplify.<\/p>\n<p>\\[<br \/>\n\\left[<br \/>\n\\begin{array}{r} <br \/>\n6 &#038; 0 &#038; 0 \\\\<br \/>\n18 &#038; 21 &#038; 0 \\\\<br \/>\n-27 &#038; 26 &#038; 42<br \/>\n\\end{array}<br \/>\n\\right]<br \/>\n\\]<\/p>\n<p>It\u2019s as easy as that! Simply take your matrix and multiply by the scalar, which in this case is 3.<\/p>\n<h3><span id=\"Multiplying_by_Other_Matrices\" class=\"m-toc-anchor\"><\/span>Multiplying by Other Matrices<\/h3>\n<p>\nIn some cases, matrices can be multiplied by each other. The dimensions of the matrices tell us two things: 1) whether multiplication is possible, and 2) the dimension of the product.<\/p>\n<p>In order for matrix multiplication to be possible, the inner dimensions must match. In other words, the number of columns of the left matrix must match the number of rows of the right matrix.<\/p>\n<div class=\"matrixcolumn-container\">\n<div class=\"matrixleftcolumn-long\">\n\\[<br \/>\nA =<br \/>\n\\left[<br \/>\n\\begin{array}{rr} <br \/>\n-1 &#038; 5 \\\\<br \/>\n6 &#038; -3 \\\\<br \/>\n7 &#038; 2<br \/>\n\\end{array}<br \/>\n\\right]<br \/>\n\\]\n<\/div>\n<div class=\"matrixrightcolumn-long\">\n\\[<br \/>\nD =<br \/>\n\\left[<br \/>\n\\begin{array}{r} <br \/>\n2 &#038; 0 &#038; 0 \\\\<br \/>\n6  &#038; 7 &#038; 0 \\\\<br \/>\n-9  &#038; 12 &#038; 14<br \/>\n\\end{array}<br \/>\n\\right]<br \/>\n\\]\n<\/div>\n<\/div>\n<p>\n&nbsp;<br \/>\nSuppose we wanted to multiply [A] times [D]. [A] is a 3\u00d72 matrix and [D] is a 3\u00d73 matrix:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/03\/Inner-Dimensions-Dont-Match.svg\" alt=\"\" width=\"125\" height=\"91\" class=\"aligncenter size-full wp-image-250243\"  role=\"img\" \/><\/p>\n<p>Since the inner dimensions don\u2019t match, we cannot multiply [A]\u00d7[D]. However, we can multiply [D]\u00d7[A]. You\u2019ll get a 3\u00d73 matrix times a 3\u00d72 matrix, so the inner dimensions do match:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/03\/Inner-Dimensions-Match.svg\" alt=\"\" width=\"125\" height=\"91\" class=\"aligncenter size-full wp-image-250246\"  role=\"img\" \/><\/p>\n<p>The number of columns of [D] matches the number of rows of [A]. Furthermore, the outer dimensions specify the dimensions of the product, so the dimension of [D]\u00d7[A] will be a 3&#215;2 matrix.<\/p>\n<p>Before we actually do the multiplication, it\u2019s important to notice a major concept here. With numbers, multiplication commutes: \\(3\\times 4 = 4\\times 3 = 12\\). With matrices, multiplication does not commute. The order of the matrices matters. As we just saw, [A]\u00d7[D]\u2260 [D]\u00d7[A].<\/p>\n<p>Here\u2019s how matrix multiplication works:<\/p>\n<div class=\"longmatrix-container\">\n\\[<br \/>\n[D][A] =<br \/>\n\\begin{bmatrix}<br \/>\nD_{1,1} &#038; D_{1,2} &#038; D_{1,3} \\\\<br \/>\nD_{2,1} &#038; D_{2,2} &#038; D_{2,3} \\\\<br \/>\nD_{3,1} &#038; D_{3,2} &#038; D_{3,3}<br \/>\n\\end{bmatrix}<br \/>\n\\begin{bmatrix}<br \/>\nA_{1,1} &#038; A_{1,2} \\\\<br \/>\nA_{2,1} &#038; A_{2,2} \\\\<br \/>\nA_{3,1} &#038; A_{3,2}<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>\\[<br \/>\n\\begin{bmatrix}<br \/>\nD_{1,1} \\times A_{1,1} + D_{1,2} \\times A_{2,1} + D_{1,3} \\times A_{3,1} &#038; D_{1,1} \\times A_{1,2} + D_{1,2} \\times A_{2,2} + D_{1,3} \\times A_{3,2} \\\\<br \/>\nD_{2,1} \\times A_{1,1} + D_{2,2} \\times A_{2,1} + D_{2,3} \\times A_{3,1} &#038; D_{2,1} \\times A_{1,2} + D_{2,2} \\times A_{2,2} + D_{2,3} \\times A_{3,2} \\\\<br \/>\nD_{3,1} \\times A_{1,1} + D_{3,2} \\times A_{2,1} + D_{3,3} \\times A_{3,1} &#038; D_{3,1} \\times A_{1,2} + D_{3,2} \\times A_{2,2} + D_{3,3} \\times A_{3,2}<br \/>\n\\end{bmatrix}<br \/>\n\\]\n<\/div>\n<p>\nThe pattern to notice here is that we move across the rows of Matrix D and down the columns of Matrix A.<\/p>\n<p>So we multiply element 1 of Matrix D row 1 times element 1 of Matrix A column 1, then add element 2 of [D] row 1 multiplied by element 2 of [A] column 1 then add element 3 of [D] row 1 multiplied by element 3 of [A] column 1. The result of row 1 of [D] times column 1 of [A] becomes row 1, column 1 of the product.<\/p>\n<p>\\[<br \/>\n[D][A] =<br \/>\n\\left[<br \/>\n\\begin{array}{r} <br \/>\n2 &#038; 0 &#038; 0 \\\\<br \/>\n6 &#038; 7 &#038; 0 \\\\<br \/>\n-9 &#038; 12 &#038; 14<br \/>\n\\end{array}<br \/>\n\\right]<br \/>\n\\left[<br \/>\n\\begin{array}{rr}<br \/>\n-1 &#038; 5 \\\\<br \/>\n6 &#038; -3 \\\\<br \/>\n7 &#038; 2<br \/>\n\\end{array}<br \/>\n\\right]<br \/>\n\\]<\/p>\n<div class=\"longmatrix-container\">\n\\[<br \/>\n=<br \/>\n\\begin{bmatrix}<br \/>\n(-2 + 0 + 0) &#038; (10 + 0 + 0) \\\\<br \/>\n(-6 + 42 + 0) &#038; (30 &#8211; 21 + 0) \\\\<br \/>\n(9 + 72 + 98) &#038; (-45 &#8211; 36 + 28)<br \/>\n\\end{bmatrix}<br \/>\n=<br \/>\n\\begin{bmatrix}<br \/>\n-2 &#038; 10 \\\\<br \/>\n36 &#038; 9 \\\\<br \/>\n179 &#038; -53<br \/>\n\\end{bmatrix}<br \/>\n\\]\n<\/div>\n<p>Now here are a few other properties to make note of. We\u2019ll use [D] to illustrate:<\/p>\n<ul>\n<li style=\"margin-bottom: 1em;\">\\(0_{3&#215;3}+D=[D]\\)<br \/>\nThe zero matrix acts like the number 0 when adding numbers \u2013 the additive identity. A number plus 0 always equals itself.<\/li>\n<li style=\"margin-bottom: 1em;\">\\(0_{3&#215;3} \\times D=D \\times 0_{3&#215;3}=0_{3&#215;3}\\)<br \/>\nThe zero matrix acts like the number 0 when multiplying numbers. A number times 0 equals 0.<\/li>\n<li>\\([D]\\cdot I_{3} = I_{3}\\cdot[D] = [D]\\)<br \/>\nThe identity matrix acts like the number 1 does in multiplication of numbers \u2013 it\u2019s the multiplicative identity. A number times 1 equals itself.<\/li>\n<\/ul>\n<p>Thanks for watching, and happy studying!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Matrices_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Matrices &#8211; Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nMatrices are commonly classified by their _________. <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\">factors<\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-2\">dimensions<\/div><div class=\"PQ\"  id=\"PQ-1-3\">lengths<\/div><div class=\"PQ\"  id=\"PQ-1-4\">width<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Matrices are generally classified by their dimensions. For example, a matrix with three rows and four columns would be considered a \u201cthree by four matrix\u201d. <\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich elements are on the main diagonal of the following matrix?<\/p>\n<p style=\"text-align: center\">\\(\\begin{bmatrix}3&#038;4&#038;7\\\\2&#038;0&#038;2\\\\0&#038;8&#038;1\\end{bmatrix}\\)<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">\\(\\begin{bmatrix}3&#038;4&#038;\\bf7\\\\2&#038;\\bf0&#038;2\\\\\\bf0&#038;8&#038;1\\end{bmatrix}\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-2\">\\(\\begin{bmatrix}\\bf3&#038;4&#038;7\\\\2&#038;\\bf0&#038;2\\\\0&#038;8&#038;\\bf1\\end{bmatrix}\\)<\/div><div class=\"PQ\"  id=\"PQ-2-3\">\\(\\begin{bmatrix}3&#038;4&#038;7\\\\2&#038;\\bf0&#038;2\\\\0&#038;8&#038;1\\end{bmatrix}\\)<\/div><div class=\"PQ\"  id=\"PQ-2-4\">\\(\\begin{bmatrix}3&#038;4&#038;7\\\\\\bf2&#038;\\bf0&#038;\\bf2\\\\0&#038;8&#038;1\\end{bmatrix}\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The <strong>main diagonal<\/strong> of a square matrix consists of elements starting from the top left corner and extending to the bottom right corner. In this example, the elements 3, 0, and 1 are on the main diagonal.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nFind the sum of matrix A and matrix B: <\/p>\n<p style=\"text-align: center\">\\(A= \\begin{bmatrix}2&#038;9&#038;3\\\\1&#038;0&#038;1\\\\2&#038;4&#038;3\\end{bmatrix}\\)<\/p>\n<p style=\"text-align: center\">\\(B= \\begin{bmatrix}6&#038;4&#038;2\\\\9&#038;9&#038;7\\\\2&#038;1&#038;0\\end{bmatrix}\\)<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">\\(\\begin{bmatrix}8&#038;0&#038;5\\\\10&#038;9&#038;4\\\\2&#038;34&#038;9\\end{bmatrix}\\)<\/div><div class=\"PQ\"  id=\"PQ-3-2\">\\(\\begin{bmatrix}3&#038;14&#038;2\\\\12&#038;24&#038;1\\\\4&#038;5&#038;3\\end{bmatrix}\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-3\">\\(\\begin{bmatrix}8&#038;13&#038;5\\\\10&#038;9&#038;8\\\\4&#038;5&#038;3\\end{bmatrix}\\)<\/div><div class=\"PQ\"  id=\"PQ-3-4\">\\(\\begin{bmatrix}8&#038;5&#038;13\\\\10&#038;9&#038;9\\\\4&#038;5&#038;3\\end{bmatrix}\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Matrix A and B can be added because they have the same dimensions. Both matrices have three rows and three columns. The corresponding elements in each matrix can be added in order to find the sum of [A] + [B]. <\/p>\n<div class=\"longmath\" style=\"text-align: center; margin-bottom: 1.5em\">\n\\(\\begin{bmatrix}2+6&#038;9+4&#038;3+2\\\\1+9&#038;0+9&#038;1+7\\\\2+2&#038;4+1&#038;3+0\\end{bmatrix}=\\begin{bmatrix}8&#038;13&#038;5\\\\10&#038;9&#038;8\\\\4&#038;5&#038;3\\end{bmatrix}\\)\n<\/div>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nIn matrices, multiplication does not ________. <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">associate<\/div><div class=\"PQ\"  id=\"PQ-4-2\">transpose<\/div><div class=\"PQ\"  id=\"PQ-4-3\">distribute<\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-4\">commute<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>When numbers are multiplied, such as \\(6\\times5\\), the numbers are able to \u201ccommute,\u201d or change positions, without affecting the result. For example:<\/p>\n<p style=\"text-align: center\">\\(6\\times5=5\\times6\\)<\/p>\n<p>However, in matrices, the order matters when multiplying. \\([A] \\times [B]\\) will not always be equal to \\([B] \\times [A]\\). If the matrices commute, the product may change.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nIf \\([R\\) is multiplied by a scalar of 3, what will the new \\([R]\\) be?<\/p>\n<p style=\"text-align: center\">\\(R= \\begin{bmatrix}5&#038;3&#038;7\\\\2&#038;3&#038;4\\\\1&#038;0&#038;9\\end{bmatrix}\\)<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-5-1\">\\(\\begin{bmatrix}15&#038;9&#038;21\\\\6&#038;9&#038;12\\\\3&#038;0&#038;27\\end{bmatrix}\\)<\/div><div class=\"PQ\"  id=\"PQ-5-2\">\\(\\begin{bmatrix}15&#038;9&#038;21\\\\12&#038;9&#038;6\\\\0&#038;3&#038;27\\end{bmatrix}\\)<\/div><div class=\"PQ\"  id=\"PQ-5-3\">\\(\\begin{bmatrix}9&#038;21&#038;15\\\\4&#038;3&#038;18\\\\3&#038;0&#038;27\\end{bmatrix}\\)<\/div><div class=\"PQ\"  id=\"PQ-5-4\">\\(\\begin{bmatrix}9&#038;15&#038;21\\\\3&#038;0&#038;12\\\\3&#038;0&#038;27\\end{bmatrix}\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>When a matrix is multiplied by a scalar, each element in the matrix is multiplied by that number. In this example, each element is multiplied by 3 to create the new matrix [R].<\/p>\n<div class=\"longmath\" style=\"text-align: center; margin-bottom: 1.5em\">\n\\(3\\begin{bmatrix}5&#038;3&#038;7\\\\2&#038;3&#038;4\\\\1&#038;0&#038;9\\end{bmatrix}=\\begin{bmatrix}3\\times5&#038;3\\times3&#038;3\\times7\\\\3\\times2&#038;3\\times3&#038;3\\times4\\\\3\\times1&#038;3\\times0&#038;3\\times9\\end{bmatrix}=\\begin{bmatrix}15&#038;9&#038;21\\\\6&#038;9&#038;12\\\\3&#038;0&#038;27\\end{bmatrix}\\)\n<\/div>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/algebra-ii\/\">Return to Algebra II Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Algebra II Videos<\/p>\n","protected":false},"author":1,"featured_media":100660,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-61701","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-math-advertising-group","7":"page_category-matrices-videos","8":"page_type-video","9":"content_type-practice-questions","10":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/61701","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=61701"}],"version-history":[{"count":6,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/61701\/revisions"}],"predecessor-version":[{"id":280400,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/61701\/revisions\/280400"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/100660"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=61701"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}