{"id":59253,"date":"2020-06-02T15:41:01","date_gmt":"2020-06-02T15:41:01","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=59253"},"modified":"2026-03-25T10:35:37","modified_gmt":"2026-03-25T15:35:37","slug":"matrices-transposition-determinants-and-augmentation","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/matrices-transposition-determinants-and-augmentation\/","title":{"rendered":"Matrices: Transposition, Determinants, and Augmentation"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_djjMGzEtRmA\" 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_djjMGzEtRmA\" data-source-videoID=\"djjMGzEtRmA\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Matrices: Transposition, Determinants, and Augmentation Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Matrices: Transposition, Determinants, and Augmentation\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_djjMGzEtRmA:hover {cursor:pointer;} img#videoThumbnailImage_djjMGzEtRmA {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/1683-matrices-transportation-determinants-and-augmentation-2.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_djjMGzEtRmA\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_djjMGzEtRmA\" 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: Transposition, Determinants, and Augmentation\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_djjMGzEtRmA\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_djjMGzEtRmA\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_djjMGzEtRmA\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction noF_Function() {\n  var x = document.getElementById(\"noF\");\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=\"noF_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=\"noF\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Transposing_Matrices\" class=\"smooth-scroll\">Transposing Matrices<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Calculating_Determinants\" class=\"smooth-scroll\">Calculating Determinants<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Augmenting_Matrices\" class=\"smooth-scroll\">Augmenting Matrices<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Matrix_Practice_Questions\" class=\"smooth-scroll\">Matrix 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 <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/matrices-data-systems\/\">matrix<\/a> manipulation! In this video, we will cover:<\/p>\n<ul>\n<li>Transposing matrices<\/li>\n<li>Finding determinants of matrices<\/li>\n<li>Augmenting matrices<\/li>\n<\/ul>\n<p>A quick preface before we get started: Generally speaking, the larger the matrices you are working with, the more tedious the work becomes. On one hand, it can be great practice to get a better conceptual understanding about what matrices do, since they are part of an interesting branch of mathematics.<\/p>\n<p>But on the other hand, once the size exceeds 3\u00d73 or so, the amount of time and paper required for some of the operations by hand is quite large. Technology of some sort is definitely recommended.<\/p>\n<h2><span id=\"Transposing_Matrices\" class=\"m-toc-anchor\"><\/span>Transposing Matrices<\/h2>\n<p>\nOne way matrices can be manipulated is to transpose them. The transpose of a matrix is denoted by a<sup>T<\/sup>. So the transpose of [A] is [A]<sup>T<\/sup>.<\/p>\n<p>To transpose a matrix, reflect all the elements over the main diagonal. In other words, row 1 of the original becomes column 1 of the transposed matrix, row 2 of the original becomes column 2 of the transposed matrix, and so on.<\/p>\n<p>You will transpose most often with square matrices. Let\u2019s look at a couple of examples:<\/p>\n<h3><span id=\"Example_1_1\" class=\"m-toc-anchor\"><\/span>Example #1<\/h3>\n<p>\nHere&#8217;s [A]:<\/p>\n<p>\\[<br \/>\n\\left[ A \\right] = <br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 2 \\\\<br \/>\n3 &#038; 4<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>If we want to find the transpose of matrix A here&#8217;s how we&#8217;d do that. The transpose of matrix A is also going to be a 2\u00d72 matrix. So we start with the first element\u2014that one&#8217;s going to stay the same. Then, from there all we&#8217;re going to do is flip our elements over the diagonal.<\/p>\n<p>With 2 and 3, we&#8217;re going to flip them (2 becomes 3, and 3 becomes 2). Then, 4 is going to stay the same.<\/p>\n<p>\\[<br \/>\n\\left[ A \\right]^T = <br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 3 \\\\<br \/>\n2 &#038; 4<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>That&#8217;s our transposed [A]!<\/p>\n<h3><span id=\"Example_2_1\" class=\"m-toc-anchor\"><\/span>Example #2<\/h3>\n<p>\nNow if we want to transpose [B], we&#8217;ll do something similar.<\/p>\n<p>\\[<br \/>\n\\left[ B \\right] = <br \/>\n\\begin{bmatrix}<br \/>\n-1 &#038; 3 &#038; -5 \\\\<br \/>\n-4 &#038; 7 &#038; 1 \\\\<br \/>\n0 &#038; 2 &#038; 9<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>So the transpose of matrix B is going to be another 3\u00d73 matrix. Remember the first element stays the same, -1. Then the next diagonal flips, so 3 and -4 are going to flip: 3 becomes -4, and -4 becomes 3.<\/p>\n<p>Now for our main diagonal we&#8217;re going to do the same thing; our three elements are going to flip. So -5 is going to go down to where the 0 is\u2014so -5 becomes 0, 7 stays the same, and 0 becomes -5.<\/p>\n<p>One more time let\u2019s now flip these two elements; 1 becomes 2, 2 becomes 1, and 9 stays the same.<\/p>\n<p>\\[<br \/>\n\\left[ B \\right]^T = <br \/>\n\\begin{bmatrix}<br \/>\n-1 &#038; -4 &#038; 0 \\\\<br \/>\n3 &#038; 7 &#038; 2 \\\\<br \/>\n-5 &#038; 1 &#038; 9<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>That&#8217;s our transposed [B]!<\/p>\n<h3><span id=\"Example_3\" class=\"m-toc-anchor\"><\/span>Example #3<\/h3>\n<p>\nHere&#8217;s [C]:<\/p>\n<p>\\[<br \/>\n\\left[ C \\right] = <br \/>\n\\begin{bmatrix}<br \/>\n9 &#038; 8 &#038; 7 \\\\<br \/>\n4 &#038; 5 &#038; 6<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>To find the transpose of [C], it&#8217;s going to look just a teeny bit different from what we did last time.<\/p>\n<p>Remember we want our first row to be our first column now. So instead of 9, 8, and 7 being a row, it&#8217;s going to be our column. Then, our second row (4, 5, 6) is going to become our second column.<\/p>\n<p>\\[<br \/>\n\\left[ C \\right]^T = <br \/>\n\\begin{bmatrix}<br \/>\n9 &#038; 4 \\\\<br \/>\n8 &#038; 5 \\\\<br \/>\n7 &#038; 6<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>That&#8217;s all there is to it!<\/p>\n<h3><span id=\"Example_4_Equal_Matrix\" class=\"m-toc-anchor\"><\/span>Example #4: Equal Matrix<\/h3>\n<p>\nSometimes, transposing a matrix yields an equal matrix.<\/p>\n<p>Consider [M] and [N] here:<\/p>\n<div class=\"longmatrix-container\">\n\\[<br \/>\n\\left[ M \\right] = <br \/>\n\\begin{bmatrix}<br \/>\n3 &#038; 7 \\\\<br \/>\n7 &#038; 2<br \/>\n\\end{bmatrix}<br \/>\n\\quad \\text{and} \\quad<br \/>\n\\left[ M \\right]^T = <br \/>\n\\begin{bmatrix}<br \/>\n3 &#038; 7 \\\\<br \/>\n7 &#038; 2<br \/>\n\\end{bmatrix}<br \/>\n\\]\n<\/div>\n<div class=\"longmatrix-container\">\n\\[<br \/>\n\\left[ N \\right] = <br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 3 &#038; 2 \\\\<br \/>\n3 &#038; 5 &#038; 6 \\\\<br \/>\n2 &#038; 6 &#038; 8<br \/>\n\\end{bmatrix}<br \/>\n\\quad \\text{and} \\quad<br \/>\n\\left[ N \\right]^T = <br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 3 &#038; 2 \\\\<br \/>\n3 &#038; 5 &#038; 6 \\\\<br \/>\n2 &#038; 6 &#038; 8<br \/>\n\\end{bmatrix}<br \/>\n\\]\n<\/div>\n<p>[M] and [N] are <strong>symmetric<\/strong>: [M]=[M]<sup>T<\/sup> and [N]=[N]<sup>T<\/sup>.<\/p>\n<h2><span id=\"Calculating_Determinants\" class=\"m-toc-anchor\"><\/span>Calculating Determinants<\/h2>\n<p>\nAnother thing we can do with matrices is calculate their determinants. Determinants, which are always a single number, will only be found when dealing with square matrices. The determinant of [A] can be denoted as det(A) or |A|.<\/p>\n<h3><span id=\"Example_1_1\" class=\"m-toc-anchor\"><\/span>Example #1<\/h3>\n<p>\nFor a 2\u00d72 matrix, the calculation of the determinant is \\(ad-bc\\).<\/p>\n<p>For example, let&#8217;s look at [A].<\/p>\n<p>If \\( \\left[ A \\right]^T = \\begin{bmatrix}1 &#038; 3 \\\\ 2 &#038; 4\\end{bmatrix} \\), then \\(det(A)\\)\\( = (1)(4) &#8211; (3)(2)\\)\\( = 4 &#8211; 6 = -2\\).<\/p>\n<p>So the determinant of [A] is -2.<\/p>\n<h3><span id=\"Example_2_1\" class=\"m-toc-anchor\"><\/span>Example #2<\/h3>\n<p>\nFor a 3\u00d73 matrix, the calculation of the determinant becomes significantly larger and involves determinants of some of the 2\u00d72 matrices within the larger matrix.<\/p>\n<p>\\[<br \/>\n\\begin{bmatrix}<br \/>\na &#038; b &#038; c \\\\<br \/>\nd &#038; e &#038; f \\\\<br \/>\ng &#038; h &#038; i<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>To find the determinant of a 3\u00d73 matrix, we will start by multiplying the elements on the top row by the determinants of the 2\u00d72 matrices created by using the four elements not in the original element&#8217;s row or column.<\/p>\n<p>So if we start with a, we will then multiply it by |e, f, h, i| because those are the four elements not in a&#8217;s row or column. <\/p>\n<p>\\[<br \/>\n\\left| <br \/>\n\\begin{array}{cc}<br \/>\na &#038; f \\\\<br \/>\nh &#038; i<br \/>\n\\end{array}<br \/>\n\\right|<br \/>\n\\]<\/p>\n<p>We will then alternate our signs, so since this first part is positive, we will subtract the second part of our equation. This time, we will start with b. We will multiply b by |d, f, g, i| because those are the four elements not included in b&#8217;s row or column.<\/p>\n<p>\\[<br \/>\n\\left| <br \/>\n\\begin{array}{ccc}<br \/>\na &#038; b &#038; c \\\\<br \/>\nd &#038; e &#038; f \\\\<br \/>\ng &#038; h &#038; i<br \/>\n\\end{array}<br \/>\n\\right| <br \/>\n= <br \/>\na \\left| <br \/>\n\\begin{array}{cc}<br \/>\ne &#038; f \\\\<br \/>\nh &#038; i<br \/>\n\\end{array}<br \/>\n\\right| <br \/>\n&#8211; <br \/>\nb \\left| <br \/>\n\\begin{array}{cc}<br \/>\nd &#038; f \\\\<br \/>\ng &#038; i<br \/>\n\\end{array}<br \/>\n\\right| <br \/>\n\\]<\/p>\n<p>Finally, we will add the last part of our equation. Remember, we do this because we are alternating signs: positive, negative, positive. So we have c times |d, e, g, h|.<\/p>\n<p>\\[<br \/>\na \\left| <br \/>\n\\begin{array}{cc}<br \/>\ne &#038; f \\\\<br \/>\nh &#038; i<br \/>\n\\end{array}<br \/>\n\\right|<br \/>\n&#8211; <br \/>\nb \\left| <br \/>\n\\begin{array}{cc}<br \/>\nd &#038; f \\\\<br \/>\ng &#038; i<br \/>\n\\end{array}<br \/>\n\\right|<br \/>\n+ <br \/>\nc \\left| <br \/>\n\\begin{array}{cc}<br \/>\nd &#038; e \\\\<br \/>\ng &#038; h<br \/>\n\\end{array}<br \/>\n\\right|<br \/>\n\\]<\/p>\n<p>Now we are able to simplify this even further since we already know how to find the determinants of 2\u00d72 matrices, so our final equation will look like this:<\/p>\n<div class=\"examplesentence\">\\(a(ei-fh)-b(di-fg)\\)\\(+c(dh-eg)\\)<\/div>\n<p>\n&nbsp;<br \/>\nPlugging your numbers into this equation would then give you the determinant of this 3\u00d73 matrix.<\/p>\n<p>Determinants can be found for any size square matrix, but this is an operation where access to technology can save much time. Usually, we are not as concerned with the calculation of the determinant as we are with what the determinant tells us about the matrix.<\/p>\n<h2><span id=\"Augmenting_Matrices\" class=\"m-toc-anchor\"><\/span>Augmenting Matrices<\/h2>\n<p>\nSometimes it will be necessary to augment matrices. Augmenting a matrix means to combine the columns of two separate matrices into a single matrix that can be manipulated as one. Note that the number of rows in both matrices must match in order to augment them.<\/p>\n<div class=\"longmatrix-container\">\n\\[<br \/>\n\\left[ A \\right] = <br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 2 \\\\<br \/>\n3 &#038; 4<br \/>\n\\end{bmatrix}<br \/>\n\\quad<br \/>\n\\left[ C \\right] = <br \/>\n\\begin{bmatrix}<br \/>\n7 \\\\<br \/>\n8<br \/>\n\\end{bmatrix}<br \/>\n\\quad<br \/>\n\\left[ D \\right] = <br \/>\n\\begin{bmatrix}<br \/>\n-3 &#038; -4 \\\\<br \/>\n6 &#038; 2<br \/>\n\\end{bmatrix}<br \/>\n\\]\n<\/div>\n<p>The notation for augmentation is a bar. For example, if we want to find the augment of matrix A and matrix C we would start with matrix A and copy it down. Then since we&#8217;re augmenting it with matrix C, we go up here and we see that we have 7 and 8. All we&#8217;re going to do is add that to the end of our other matrix. That gives us our augmented matrix:<\/p>\n<p>\\[<br \/>\n\\left[ A \\mid C \\right] = <br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 2 &#038; 7 \\\\<br \/>\n3 &#038; 4 &#038; 8<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>Now let&#8217;s augment matrix A and matrix D. Remember, start with A and just copy it down. Then we simply add D to the end of A.<\/p>\n<p>\\[<br \/>\n\\left[ A \\mid D \\right] = <br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 2 &#038; -3 &#038; -4 \\\\<br \/>\n3 &#038; 4 &#038; 6 &#038; 2<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>And that&#8217;s it! That&#8217;s how you augment matrices.<\/p>\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=\"Matrix_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Matrix 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 \/>\nFind the transpose of the matrix below.<\/p>\n<p style=\"text-align:center; line-height: 50px;\">\n\\(\\begin{bmatrix}<br \/>\n4&#038;5&#038;6\\\\<br \/>\n-1&#038;2&#038;9\\\\<br \/>\n4&#038;-7&#038;1<br \/>\n\\end{bmatrix}\\)\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-1-1\">\\(\\begin{bmatrix}\r\n4&#038;-1&#038;4\\\\\r\n5&#038;2&#038;-7\\\\\r\n6&#038;9&#038;1\r\n\\end{bmatrix}\\)<\/div><div class=\"PQ\"  id=\"PQ-1-2\">\\(\\begin{bmatrix}\r\n3&#038;-1&#038;3\\\\\r\n5&#038;1&#038;-7\\\\\r\n2&#038;9&#038;1\r\n\\end{bmatrix}\\)<\/div><div class=\"PQ\"  id=\"PQ-1-3\">\\(\\begin{bmatrix}\r\n4&#038;-1&#038;4\\\\\r\n1&#038;2&#038;-7\\\\\r\n6&#038;0&#038;1\r\n\\end{bmatrix}\\)<\/div><div class=\"PQ\"  id=\"PQ-1-4\">\\(\\begin{bmatrix}\r\n4&#038;-1&#038;4\\\\\r\n0&#038;2&#038;-7\\\\\r\n0&#038;0&#038;1\r\n\\end{bmatrix}\\)<\/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>To transpose the matrix, elements are reflected over the main diagonal. This means that the 5and the \u22121 will change places, the 6 and the 4 will change places, and the 9 and the \u22127 will change places. <\/p>\n<p style=\"text-align:center; line-height: 50px;\">\n\\(\\begin{bmatrix}<br \/>\n\\bf4&#038;5&#038;6\\\\<br \/>\n-1&#038;\\bf2&#038;9\\\\<br \/>\n4&#038;-7&#038;\\bf1<br \/>\n\\end{bmatrix}\\)<br \/>\nbecomes<br \/>\n\\(\\begin{bmatrix}<br \/>\n4&#038;-1&#038;4\\\\<br \/>\n5&#038;2&#038;-7\\\\<br \/>\n6&#038;9&#038;1<br \/>\n\\end{bmatrix}\\)<\/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 matrix will yield an equal matrix when transposed?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-2-1\">\\(\\begin{bmatrix}\r\n4&#038;6&#038;2\\\\\r\n6&#038;5&#038;3\\\\\r\n2&#038;3&#038;7\r\n\\end{bmatrix}\\)<\/div><div class=\"PQ\"  id=\"PQ-2-2\">\\(\\begin{bmatrix}\r\n3&#038;5&#038;7\\\\\r\n2&#038;3&#038;3\\\\\r\n5&#038;4&#038;3\r\n\\end{bmatrix}\\)<\/div><div class=\"PQ\"  id=\"PQ-2-3\">\\(\\begin{bmatrix}\r\n0&#038;0&#038;1\\\\\r\n1&#038;3&#038;0\\\\\r\n5&#038;4&#038;3\r\n\\end{bmatrix}\\)<\/div><div class=\"PQ\"  id=\"PQ-2-4\">\\(\\begin{bmatrix}\r\n3&#038;0&#038;0\\\\\r\n1&#038;7&#038;8\\\\\r\n3&#038;2&#038;6\r\n\\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 style=\"text-align:center;\">\n\\(\\begin{bmatrix}<br \/>\n\\bf4&#038;6&#038;2\\\\<br \/>\n6&#038;\\bf5&#038;3\\\\<br \/>\n2&#038;3&#038;\\bf7<br \/>\n\\end{bmatrix}\\)<\/p>\n<p>The original matrix is symmetric. This means that when the elements are reflected over the main diagonal, the new matrix is identical to the original matrix. <\/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 \/>\nCalculate the determinant of the following matrix:<\/p>\n<p style=\"text-align:center; line-height: 50px;\">\\(\\begin{bmatrix}<br \/>\n1&#038;5\\\\<br \/>\n2&#038;7<br \/>\n\\end{bmatrix}\\)<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">\u22125<\/div><div class=\"PQ\"  id=\"PQ-3-2\">\u22124<\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-3\">\u22123<\/div><div class=\"PQ\"  id=\"PQ-3-4\">\u22122<\/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>The formula \\(ad-bc\\) can be used to calculate the determinant of the matrix.<\/p>\n<p>In the original matrix, \\(a=1\\), \\(b=5\\), \\(c=2\\) , and \\(d=7\\). When these values are plugged into the formula, \\(ad-bc\\) becomes \\((1)(7)-(5)(2)\\), which simplifies to \\(7-10\\), or \u22123.<\/p>\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 \/>\nThe bar between \\(R\\) and \\(Y\\) below indicates the notation for _________________.<\/p>\n<p style=\"text-align:center;\">\n\\(\\begin{bmatrix}<br \/>\nR | Y<br \/>\n\\end{bmatrix}\\)<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">row reduction<\/div><div class=\"PQ\"  id=\"PQ-4-2\">determinants<\/div><div class=\"PQ\"  id=\"PQ-4-3\">transposition<\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-4\">augmentation<\/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>The bar symbol indicates augmentation. Augmenting a matrix means combining the columns into a single matrix that can be manipulated as one. <\/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>Augment matrix E and matrix K.<\/p>\n<p style=\"text-align:center;\">\n\\([E]\\begin{bmatrix}3&#038;5\\\\2&#038;1\\end{bmatrix}<br \/>\n\\quad<br \/>\n[K]\\begin{bmatrix}7\\\\0\\end{bmatrix}\\)<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\">\\([E|K]\\)\r\n\\(\\left[\\begin{matrix}\r\n4&#038;5\\\\\r\n2&#038;1\\end{matrix}\r\n\\left|\\,\\begin{matrix}7\\\\1\\end{matrix}\\right.\\right]\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-2\">\\([E|K]\\)\r\n\\(\\left[\\begin{matrix}\r\n3&#038;5\\\\\r\n2&#038;1\\end{matrix}\r\n\\left|\\,\\begin{matrix}7\\\\0\\end{matrix}\\right.\\right]\\)<\/div><div class=\"PQ\"  id=\"PQ-5-3\">\\([E|K]\\)\r\n\\(\\left[\\begin{matrix}\r\n7&#038;3\\\\\r\n2&#038;1\\end{matrix}\r\n\\left|\\,\\begin{matrix}5\\\\0\\end{matrix}\\right.\\right]\\)<\/div><div class=\"PQ\"  id=\"PQ-5-4\">\\([E|K]\\)\r\n\\(\\left[\\begin{matrix}\r\n2&#038;1\\\\\r\n3&#038;7\\end{matrix}\r\n\\left|\\,\\begin{matrix}0\\\\5\\end{matrix}\\right.\\right]\\)<\/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>Augmenting matrices occurs when the columns of two separate matrices are combined into a single matrix that can be manipulated. The number of rows must be the same in order for the matrices to be augmented.<\/p>\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":100663,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-59253","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\/59253","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=59253"}],"version-history":[{"count":5,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/59253\/revisions"}],"predecessor-version":[{"id":278890,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/59253\/revisions\/278890"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/100663"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=59253"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}