{"id":61508,"date":"2020-09-02T15:55:24","date_gmt":"2020-09-02T15:55:24","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=61508"},"modified":"2026-05-01T11:21:46","modified_gmt":"2026-05-01T16:21:46","slug":"matrices-geometric-transformations","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/matrices-geometric-transformations\/","title":{"rendered":"Matrices: Geometric Transformations"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_nGOl48x3Gcs\" 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_nGOl48x3Gcs\" data-source-videoID=\"nGOl48x3Gcs\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Matrices: Geometric Transformations Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Matrices: Geometric Transformations\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_nGOl48x3Gcs:hover {cursor:pointer;} img#videoThumbnailImage_nGOl48x3Gcs {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/1686-matrices-geometric-transformation-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_nGOl48x3Gcs\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_nGOl48x3Gcs\" 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: Geometric Transformations\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_nGOl48x3Gcs\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_nGOl48x3Gcs\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_nGOl48x3Gcs\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction vUH_Function() {\n  var x = document.getElementById(\"vUH\");\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=\"vUH_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=\"vUH\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Matrix_Transformations_on_the_Coordinate_Plane\" class=\"smooth-scroll\">Matrix Transformations on the Coordinate Plane<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Translations_with_Matrices\" class=\"smooth-scroll\">Translations with Matrices<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Dilations_with_Matrices\" class=\"smooth-scroll\">Dilations with Matrices<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Reflections_with_Matrices\" class=\"smooth-scroll\">Reflections with Matrices<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Rotations_with_Matrices\" class=\"smooth-scroll\">Rotations with Matrices<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Matrices_Geometric_Transformation_Practice_Questions\" class=\"smooth-scroll\">Matrices Geometric Transformation 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 using matrices to transform figures on the <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/cartesian-coordinate-plane-and-graphing\/\">coordinate plane<\/a>!<\/p>\n<p>In this video, we will cover translations, dilations, reflections, and rotations.<\/p>\n<h2><span id=\"Matrix_Transformations_on_the_Coordinate_Plane\" class=\"m-toc-anchor\"><\/span>Matrix Transformations on the Coordinate Plane<\/h2>\n<p>\nUsing addition, subtraction, scalar multiplication, and matrix multiplication, we can transform figures on the coordinate plane. All we need are the coordinates of the figure, which can be any shape and does not need to be closed. This triangle will be used to demonstrate each transformation:<\/p>\n<h3><span id=\"Creating_a_Coordinate_Matrix\" class=\"m-toc-anchor\"><\/span>Creating a Coordinate Matrix<\/h3>\n<p>\nFirst, we need to create the <strong>coordinate matrix<\/strong> for the figure. The general form of a coordinate matrix is \\(x_1\\), \\(x_2\\), \\(x_3\\), \\(y_1\\), \\(y_2\\), \\(y_3\\), and so on.<\/p>\n<p>\\[<br \/>\n\\begin{bmatrix}<br \/>\nx_1 &#038; x_2 &#038; x_3 \\\\<br \/>\ny_1 &#038; y_2 &#038; y_3<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>The key to remembering this <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/matrices-elementary-row-operations\/\">matrix<\/a> is remembering that row 1 represents the \\(x\\)-coordinates and row 2 represents the \\(y\\)-coordinates.<\/p>\n<p>The coordinate matrix for our triangle, we\u2019ll call it \\(T\\), is:<\/p>\n<p>\\[<br \/>\nT =<br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 2 &#038; 3 \\\\<br \/>\n0 &#038; 3 &#038; 1<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>Now we can perform operations on our coordinate matrix in order to discover the coordinates of our triangle transformed in various ways.<\/p>\n<h2><span id=\"Translations_with_Matrices\" class=\"m-toc-anchor\"><\/span>Translations with Matrices<\/h2>\n<h3><span id=\"Horizontal_and_Vertical_Translations\" class=\"m-toc-anchor\"><\/span>Horizontal and Vertical Translations<\/h3>\n<p>\nTranslations, or slides, can be performed simply by adding the amount and direction of the slide to the \\(x\\)&#8211; and \\(y\\)-coordinates separately.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/05\/Triangle-translation-1.svg\" alt=\"A triangle with vertices labeled A, B, and C is plotted on an x-y grid. The triangle spans from (1,1) to (2,3) to (3,1).\" width=\"444\" height=\"348\" class=\"aligncenter size-full wp-image-292826\"  role=\"img\" \/><\/p>\n<p>Suppose we wanted to slide our triangle two units to the right. Since this is a change involving the \\(x\\)-coordinates moving in the positive direction, we would simply perform a matrix addition that looks like this, which causes 2 to be added to each \\(x\\)-coordinate:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/05\/Triangle-translation-2.svg\" alt=\"Two triangles on a coordinate plane: triangle ABC at (1,1), (2,3), (3,2) and triangle A&#039;B&#039;C&#039; at (3,1), (4,3), (5,2).\" width=\"444\" height=\"348\" class=\"aligncenter size-full wp-image-292829\"  role=\"img\" \/><\/p>\n<p>\\[<br \/>\n\\begin{bmatrix}<br \/>\n2 &#038; 2 &#038; 2 \\\\<br \/>\n0 &#038; 0 &#038; 0<br \/>\n\\end{bmatrix}<br \/>\n+<br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 2 &#038; 3 \\\\<br \/>\n0 &#038; 3 &#038; 1<br \/>\n\\end{bmatrix}<br \/>\n=<br \/>\n\\begin{bmatrix}<br \/>\n3 &#038; 4 &#038; 5 \\\\<br \/>\n0 &#038; 3 &#038; 1<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>The coordinates of our translated triangle are \\((3,0)\\), \\((4,3)\\), and \\((5,1)\\), as shown on the graph.<\/p>\n<p>Pretty simple, right? Suppose we wanted to slide our triangle three units down from its original position. The matrix addition would look like this, which causes three to be subtracted from each \\(y\\)-coordinate:<\/p>\n<p>\\[<br \/>\n\\begin{bmatrix}<br \/>\n0 &#038; 0 &#038; 0 \\\\<br \/>\n-3 &#038; -3 &#038; -3<br \/>\n\\end{bmatrix}<br \/>\n+<br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 2 &#038; 3 \\\\<br \/>\n0 &#038; 3 &#038; 1<br \/>\n\\end{bmatrix}<br \/>\n=<br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 2 &#038; 3 \\\\<br \/>\n-3 &#038; 0 &#038; -2<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<h3><span id=\"Combining_Translations\" class=\"m-toc-anchor\"><\/span>Combining Translations<\/h3>\n<p>\nOf course, if we wanted to perform both transformations at once, it would look like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/05\/Triangle-translation-3.svg\" alt=\"Two blue triangles are shown on a coordinate grid, one above the x-axis and the other below, with labeled vertices A, B, C and A&#039;, B&#039;, C&#039;.\" width=\"444\" height=\"501\" class=\"aligncenter size-full wp-image-292832\"  role=\"img\" \/><\/p>\n<p>\\[<br \/>\n\\begin{bmatrix}<br \/>\n2 &#038; 2 &#038; 2 \\\\<br \/>\n-3 &#038; -3 &#038; -3<br \/>\n\\end{bmatrix}<br \/>\n+<br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 2 &#038; 3 \\\\<br \/>\n0 &#038; 3 &#038; 1<br \/>\n\\end{bmatrix}<br \/>\n=<br \/>\n\\begin{bmatrix}<br \/>\n3 &#038; 4 &#038; 5 \\\\<br \/>\n-3 &#038; 0 &#038; -2<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>We put our positive 2s in the top row to move the triangle along the \\(x\\)-axis, and put negative 3s in the bottom row to also move it along the \\(y\\)-axis.<\/p>\n<h2><span id=\"Dilations_with_Matrices\" class=\"m-toc-anchor\"><\/span>Dilations with Matrices<\/h2>\n<p>\nSuppose we wanted to make our triangle twice as large. We would simply multiply our coordinate matrix by a factor of 2, like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/05\/Triangle-translation-4.svg\" alt=\"Two triangles, one smaller and one larger, are plotted on an x-y coordinate grid, with the larger triangle appearing as an enlarged version of the smaller triangle.\" width=\"444\" height=\"447\" class=\"aligncenter size-full wp-image-292835\"  role=\"img\" \/><\/p>\n<p>\\[<br \/>\n2<br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 2 &#038; 3 \\\\<br \/>\n0 &#038; 3 &#038; 1<br \/>\n\\end{bmatrix}<br \/>\n=<br \/>\n\\begin{bmatrix}<br \/>\n2 &#038; 4 &#038; 6 \\\\<br \/>\n0 &#038; 6 &#038; 2<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>This transformation is called a <strong>dilation<\/strong>. Dilations can be either expansions or reductions. Instead of expanding our shape like we just did, we might have reduced its size by a factor of one-half like this:<\/p>\n<p>\\[<br \/>\n\\dfrac{1}{2}<br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 2 &#038; 3 \\\\<br \/>\n0 &#038; 3 &#038; 1<br \/>\n\\end{bmatrix}<br \/>\n=<br \/>\n\\begin{bmatrix}<br \/>\n\\frac{1}{2} &#038; 1 &#038; \\frac{3}{2} \\\\<br \/>\n0 &#038; \\frac{3}{2} &#038; \\frac{1}{2}<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<h2><span id=\"Reflections_with_Matrices\" class=\"m-toc-anchor\"><\/span>Reflections with Matrices<\/h2>\n<h3><span id=\"Reflecting_Over_the_Axes\" class=\"m-toc-anchor\"><\/span>Reflecting Over the Axes<\/h3>\n<p>\nWe can also use matrices to <strong>reflect<\/strong> figures in various ways. This is done with matrix multiplication.<\/p>\n<p>Let\u2019s briefly examine the big picture first. If we multiply \\(T\\) by the 2\u00d72 identity matrix, we don\u2019t change our triangle\u2019s size or position. Let\u2019s take a look:<\/p>\n<p>\\[<br \/>\nI \\times T =<br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 0 \\\\<br \/>\n0 &#038; 1<br \/>\n\\end{bmatrix}<br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 2 &#038; 3 \\\\<br \/>\n0 &#038; 3 &#038; 1<br \/>\n\\end{bmatrix}<br \/>\n=<br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 2 &#038; 3 \\\\<br \/>\n0 &#038; 3 &#038; 1<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>Essentially, the \\(x\\)-coordinates are multiplied by 1 and the \\(y\\)-coordinates are multiplied by 1, so they don\u2019t change. Remember, the identity matrix is the multiplicative identity, so this is what\u2019s supposed to happen.<\/p>\n<p>Suppose we wanted to reflect our triangle over the \\(y\\)-axis. None of the \\(y\\)-coordinates would change and all the \\(x\\)-coordinates would become the inverse. Here\u2019s what the multiplication looks like:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/05\/Triangle-translation-5.svg\" alt=\"A graph shows two congruent triangles, one on each side of the y-axis, illustrating reflection. Triangles are labeled A, B, C and A&#039;, B&#039;, C&#039;.\" width=\"441\" height=\"342\" class=\"aligncenter size-full wp-image-292841\"  role=\"img\" \/><\/p>\n<p>\\[<br \/>\n\\begin{bmatrix}<br \/>\n-1 &#038; 0 \\\\<br \/>\n0 &#038; 1<br \/>\n\\end{bmatrix}<br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 2 &#038; 3 \\\\<br \/>\n0 &#038; 3 &#038; 1<br \/>\n\\end{bmatrix}<br \/>\n=<br \/>\n\\begin{bmatrix}<br \/>\n-1 &#038; -2 &#038; -3 \\\\<br \/>\n0 &#038; 3 &#038; 1<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>All the \\(x\\)-coordinates are now opposite the originals, and our triangle has been reflected.<\/p>\n<h3><span id=\"Reflecting_Over_the_Origin\" class=\"m-toc-anchor\"><\/span>Reflecting Over the Origin<\/h3>\n<p>\nIf you wanted to reflect the triangle over the \\(x\\)-axis, you would go through the same process, but use this matrix to multiply:<\/p>\n<p>\\[<br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 0 \\\\<br \/>\n0 &#038; -1<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>The new \\(y\\)-coordinates are the inverse of the original. \\((x,y)\\) becomes \\((x,-y)\\).<\/p>\n<p>If you wanted to reflect the triangle over the origin, meaning reflect it simultaneously over both axes, you would use this matrix to multiply:<\/p>\n<p>\\[<br \/>\n\\begin{bmatrix}<br \/>\n-1 &#038; 0 \\\\<br \/>\n0 &#038; -1<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>The new \\(x\\)&#8211; and \\(y\\)-coordinates are both inverse of the originals. \\((x,y)\\) becomes \\((-x,-y)\\).<\/p>\n<h2><span id=\"Rotations_with_Matrices\" class=\"m-toc-anchor\"><\/span>Rotations with Matrices<\/h2>\n<h3><span id=\"Common_Rotation_Matrices\" class=\"m-toc-anchor\"><\/span>Common Rotation Matrices<\/h3>\n<p>\nLastly, let\u2019s look at rotation. We can <strong>rotate<\/strong> our triangle by any angle measure in the counterclockwise direction. First, let\u2019s look at some common rotations, then we\u2019ll see how to rotate any angle amount.<\/p>\n<p>Suppose we want to rotate our triangle by 90 degrees. We would simply multiply our coordinate matrix by this rotation matrix:<\/p>\n<p>\\[<br \/>\n\\begin{bmatrix}<br \/>\n0 &#038; -1 \\\\<br \/>\n1 &#038; 0<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>This gives us this matrix:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/05\/Triangle-translation-6.svg\" alt=\"Two triangles, ABC and A&#039;B&#039;C&#039;, are shown on a coordinate grid. Triangle ABC is in the first quadrant; triangle A&#039;B&#039;C&#039; is in the second quadrant.\" width=\"441\" height=\"342\" class=\"aligncenter size-full wp-image-292847\"  role=\"img\" \/><\/p>\n<p>\\[<br \/>\n\\begin{bmatrix}<br \/>\n0 &#038; -1 \\\\<br \/>\n1 &#038; 0<br \/>\n\\end{bmatrix}<br \/>\n\\begin{bmatrix}<br \/>\n1 &#038; 2 &#038; 3 \\\\<br \/>\n0 &#038; 3 &#038; 1<br \/>\n\\end{bmatrix}<br \/>\n=<br \/>\n\\begin{bmatrix}<br \/>\n0 &#038; -3 &#038; -1 \\\\<br \/>\n1 &#038; 2 &#038; 3<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>This is how the triangle is now positioned. A 180-degree rotation is the same as reflecting about the origin, so we use the same matrix.<\/p>\n<p>For a 270-degree counterclockwise rotation, we would use this matrix:<\/p>\n<p>\\[<br \/>\n\\begin{bmatrix}<br \/>\n0 &#038; 1 \\\\<br \/>\n-1 &#038; 0<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<h3><span id=\"The_General_Rotation_Matrix\" class=\"m-toc-anchor\"><\/span>The General Rotation Matrix<\/h3>\n<p>\nThere are some clear parallels to reflections here, but there is also a trigonometry connection that allows us to rotate any angle amount we want. The general matrix to use to multiply looks like this:<\/p>\n<p>\\[<br \/>\n\\begin{bmatrix}<br \/>\n\\cos\\theta &#038; -\\sin\\theta \\\\<br \/>\n\\sin\\theta &#038; \\cos\\theta<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>We can quickly see where our three common matrices come from:<\/p>\n<p>A 90-degree counterclockwise rotation would look like this:<\/p>\n<p>\\[<br \/>\n\\begin{bmatrix}<br \/>\n\\cos90 &#038; -\\sin90 \\\\<br \/>\n\\sin90 &#038; \\cos90<br \/>\n\\end{bmatrix}<br \/>\n=<br \/>\n\\begin{bmatrix}<br \/>\n0 &#038; -1 \\\\<br \/>\n1 &#038; 0<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>A 180-degree counterclockwise rotation might look like this:<\/p>\n<p>\\[<br \/>\n\\begin{bmatrix}<br \/>\n\\cos180 &#038; -\\sin180 \\\\<br \/>\n\\sin180 &#038; \\cos180<br \/>\n\\end{bmatrix}<br \/>\n=<br \/>\n\\begin{bmatrix}<br \/>\n-1 &#038; 0 \\\\<br \/>\n0 &#038; -1<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>And finally, a 270-degree counterclockwise rotation looks like this:<\/p>\n<p>\\[<br \/>\n\\begin{bmatrix}<br \/>\n\\cos270 &#038; -\\sin270 \\\\<br \/>\n\\sin270 &#038; \\cos270<br \/>\n\\end{bmatrix}<br \/>\n=<br \/>\n\\begin{bmatrix}<br \/>\n0 &#038; 1 \\\\<br \/>\n-1 &#038; 0<br \/>\n\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>But now we can rotate by other angles, like 30 or 45 degrees if we want to.<\/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=\"Matrices_Geometric_Transformation_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Matrices Geometric Transformation 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 \/>\nThe coordinates of the vertices of quadrilateral \\(Q\\) are shown in the matrix below:<\/p>\n<p style=\"text-align:center;\">\\(Q=\\begin{bmatrix}<br \/>\n3 &#038; 1 &#038; 2 &#038; 4\\\\<br \/>\n-3 &#038; 0 &#038; 0 &#038; -2<br \/>\n\\end{bmatrix}<br \/>\n\\)<\/p>\n<p>What are the coordinates of the vertices of quadrilateral \\(Q\\) after it has been translated 4 units left and 3 units up?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-1-1\">\\(\\begin{bmatrix}\r\n-1 &#038; -3 &#038; -2 &#038; 0\\\\ \r\n0 &#038; 3 &#038; 3 &#038; 1\r\n\\end{bmatrix}\r\n\\)<\/div><div class=\"PQ\"  id=\"PQ-1-2\">\\(\\begin{bmatrix}\r\n7 &#038; 5 &#038; 6 &#038; 8\\\\ \r\n0 &#038; 3 &#038; 3 &#038; 1\r\n\\end{bmatrix}\r\n\\)<\/div><div class=\"PQ\"  id=\"PQ-1-3\">\\(\\begin{bmatrix}\r\n-1 &#038; -3 &#038; -2 &#038; 0\\\\ \r\n-6 &#038; -3 &#038; -3 &#038; -5\r\n\\end{bmatrix}\r\n\\)<\/div><div class=\"PQ\"  id=\"PQ-1-4\">\\(\\begin{bmatrix}\r\n7 &#038; 5 &#038; 6 &#038; 8\\\\ \r\n-6 &#038; -3 &#038; -3 &#038; -5\r\n\\end{bmatrix}\r\n\\)<\/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 translate the triangle 4 units left and 3 units up we will add the original matrix to the matrix reflecting the translation:<\/p>\n<p class=\"longmath\" style=\"text-align: center\">\n\\(\\begin{bmatrix}<br \/>\n-4 &#038; -4 &#038; -4 &#038;-4 \\\\<br \/>\n 3&#038; 3 &#038; 3 &#038; 3<br \/>\n\\end{bmatrix}+\\begin{bmatrix}<br \/>\n3 &#038; 1 &#038; 2 &#038; 4\\\\<br \/>\n-3 &#038; 0 &#038; 0 &#038; -2<br \/>\n\\end{bmatrix}\\)<\/p>\n<p style=\"text-align: center\">\\(=\\begin{bmatrix}<br \/>\n-1 &#038; -3 &#038; -2 &#038; 0\\\\<br \/>\n0 &#038; 3 &#038; 3 &#038; 1<br \/>\n\\end{bmatrix}<br \/>\n\\)<\/p>\n<p>Therefore, the coordinates of the vertices of quadrilateral \\(Q\\) after it has been translated 4 units up and 3 units left are:<\/p>\n<p style=\"text-align:center;\">\\(\\begin{bmatrix}<br \/>\n-1 &#038; -3 &#038; -2 &#038; 0\\\\<br \/>\n0 &#038; 3 &#038; 3 &#038; 1<br \/>\n\\end{bmatrix}<br \/>\n\\)<\/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 \/>\n\\([K]\\) below shows the coordinates of the vertices of a triangle.<\/p>\n<p style=\"text-align: center;\">\\(K=\\begin{bmatrix}<br \/>\n-1 &#038; 2 &#038;-1 \\\\<br \/>\n 3&#038; 4 &#038; 1<br \/>\n\\end{bmatrix}<br \/>\n\\)<\/p>\n<p>What are the coordinates of the vertices of \\(\\triangle K\\) after a dilation by a factor of 3?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">\\(\\begin{bmatrix}\r\n2 &#038; 5 &#038; 2\\\\ \r\n 6&#038; 7 &#038; 4\r\n\\end{bmatrix}\r\n\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-2\">\\(\\begin{bmatrix}\r\n-3 &#038; 6 &#038; -3\\\\ \r\n 9&#038; 12 &#038; 3\r\n\\end{bmatrix}\r\n\\)<\/div><div class=\"PQ\"  id=\"PQ-2-3\">\\(\\begin{bmatrix}\r\n4 &#038; 1 &#038; 4\\\\ \r\n 0&#038; -1 &#038; 2\r\n\\end{bmatrix}\r\n\\)<\/div><div class=\"PQ\"  id=\"PQ-2-4\">\\(\\begin{bmatrix}\r\n3 &#038; -6 &#038; 3\\\\ \r\n -9&#038; -12 &#038; -3\r\n\\end{bmatrix}\r\n\\)<\/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>To dilate by a factor of 3, we must multiply all the entries in the matrix by 3, which will result in the following matrix:<\/p>\n<p style=\"text-align: center\">\\(\\begin{bmatrix}<br \/>\n-3 &#038; 6 &#038; -3\\\\<br \/>\n 9&#038; 12 &#038; 3<br \/>\n\\end{bmatrix}<br \/>\n\\)<\/p>\n<p>These are the coordinates of \\(\\triangle K\\) after it has been dilated by a factor of 3.<\/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 \/>\nThe vertices of the coordinates of \\(\\triangle W\\) are shown in the matrix below.<\/p>\n<p style=\"text-align: center;\">\\(W=\\begin{bmatrix}<br \/>\n-4 &#038; -1 &#038; 0\\\\<br \/>\n-1 &#038; 1 &#038; -2<br \/>\n\\end{bmatrix}<br \/>\n\\)<\/p>\n<p>What are the coordinates of the vertices of \\(\\triangle W\\) after the size is reduced to \\(\\large{\\frac{1}{3}}\\) of the original size?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">\\(\\begin{bmatrix}\r\n-7 &#038; -4 &#038; -3\\\\ \r\n-4 &#038; -2 &#038; -5\r\n\\end{bmatrix}\r\n\\)<\/div><div class=\"PQ\"  id=\"PQ-3-2\">\\(\\begin{bmatrix}\r\n-1 &#038; 2 &#038; 3\\\\ \r\n2 &#038; 4 &#038; 1\r\n\\end{bmatrix}\r\n\\)<\/div><div class=\"PQ\"  id=\"PQ-3-3\">\\(\\begin{bmatrix}\r\n\\frac{4}{3} &#038; \\frac{1}{3} &#038; 0\\\\ \r\n\\frac{1}{3} &#038; -\\frac{1}{3} &#038; \\frac{2}{3}\r\n\\end{bmatrix}\r\n\\)<\/div><div class=\"PQ\"  id=\"PQ-3-4\">\\(\\begin{bmatrix}\r\n-\\frac{4}{3} &#038; -\\frac{1}{3} &#038; 0\\\\ \r\n-\\frac{1}{3} &#038; \\frac{1}{3} &#038; -\\frac{2}{3}\r\n\\end{bmatrix}\r\n\\)<\/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>To reduce the size of a triangle, multiply the original coordinates by the factor it is being reduced by (in this case \\(\\frac{1}{3}\\)), which results in the following coordinates for the vertices:<\/p>\n<p style=\"text-align:center;\">\\(\\begin{bmatrix}<br \/>\n-\\frac{4}{3} &#038; -\\frac{1}{3} &#038; 0\\\\<br \/>\n-\\frac{1}{3} &#038; \\frac{1}{3} &#038; -\\frac{2}{3}<br \/>\n\\end{bmatrix}<br \/>\n\\)<\/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 coordinates of the vertices of quadrilateral \\(L\\) are shown in the following matrix:<\/p>\n<p style=\"text-align:center;\">\\(L=\\begin{bmatrix}<br \/>\n0 &#038; 3 &#038; 2 &#038; -3\\\\<br \/>\n1 &#038; 3 &#038; 4 &#038; 5<br \/>\n\\end{bmatrix}<br \/>\n\\)<\/p>\n<p>What are the coordinates of the vertices of quadrilateral \\(L\\) after it has been reflected across the \\(y\\)-axis?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">\\(\\begin{bmatrix}\r\n0 &#038; 3 &#038; 2 &#038; -3\\\\ \r\n-1 &#038; -3 &#038; -4 &#038; -5\r\n\\end{bmatrix}\r\n\\)<\/div><div class=\"PQ\"  id=\"PQ-4-2\">\\(\\begin{bmatrix}\r\n0 &#038; -3 &#038; -2 &#038; 3\\\\ \r\n-1 &#038; -3 &#038; -4 &#038; -5\r\n\\end{bmatrix}\r\n\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-3\">\\(\\begin{bmatrix}\r\n0 &#038; -3 &#038; -2 &#038; 3\\\\ \r\n1 &#038; 3 &#038; 4 &#038; 5\r\n\\end{bmatrix}\r\n\\)<\/div><div class=\"PQ\"  id=\"PQ-4-4\">\\(\\begin{bmatrix}\r\n1 &#038; 3 &#038; 4 &#038; 5\\\\ \r\n0 &#038; 3 &#038; 2 &#038; -3\r\n\\end{bmatrix}\r\n\\)<\/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>We can find the coordinates of the vertices of quadrilateral \\(L\\) after a reflection across the \\(y\\)-axis by multiplying the matrix by the following:<\/p>\n<p style=\"text-align: center\">\\(\\begin{bmatrix}<br \/>\n-1 &#038; 0\\\\<br \/>\n0 &#038; 1<br \/>\n\\end{bmatrix}\\)<\/p>\n<p>This results in the value of the \\(y\\)-coordinates staying the same and the \\(x\\)-values changing sign:<\/p>\n<p style=\"text-align: center;\">\\(\\begin{bmatrix}<br \/>\n0 &#038; -3 &#038; -2 &#038; 3\\\\<br \/>\n1 &#038; 3 &#038; 4 &#038; 5<br \/>\n\\end{bmatrix}<br \/>\n\\)<\/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 \/>\nThe coordinates of the vertices of \\(\\triangle T\\) are shown in the following matrix:<\/p>\n<p style=\"text-align:center;\">\\(T=\\begin{bmatrix}<br \/>\n2 &#038; 5 &#038; 3\\\\<br \/>\n-3 &#038; 1 &#038; 0<br \/>\n\\end{bmatrix}<br \/>\n\\)<\/p>\n<p>What are the coordinates of the vertices of \\(\\triangle T\\) after it has been rotated 90\u00b0 counterclockwise?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-5-1\">\\(\\begin{bmatrix}\r\n3 &#038; -1 &#038; 0\\\\ \r\n2 &#038; 5 &#038; 3\r\n\\end{bmatrix}\r\n\\)<\/div><div class=\"PQ\"  id=\"PQ-5-2\"> \\(\\begin{bmatrix}\r\n-3 &#038; 1 &#038; 0\\\\ \r\n2 &#038; 5 &#038; 3\r\n\\end{bmatrix}\r\n\\)<\/div><div class=\"PQ\"  id=\"PQ-5-3\">\\(\\begin{bmatrix}\r\n-3 &#038; 1 &#038; 0\\\\ \r\n-2 &#038; -5 &#038; -3\r\n\\end{bmatrix}\r\n\\)<\/div><div class=\"PQ\"  id=\"PQ-5-4\">\\(\\begin{bmatrix}\r\n3 &#038; -1 &#038; 0\\\\ \r\n-2 &#038; -5 &#038; -3\r\n\\end{bmatrix}\r\n\\)<\/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>To use matrices to rotate the coordinates of the vertices of \\(\\triangle T\\) by 90\u00b0, multiply the coordinates of the vertices of \\(\\triangle T\\) by the following:<\/p>\n<p style=\"text-align: center\">\\(\\begin{bmatrix}<br \/>\n0 &#038; -1\\\\<br \/>\n1 &#038; 0<br \/>\n\\end{bmatrix}<br \/>\n\\)<\/p>\n<p>This results in the following coordinates:<\/p>\n<p style=\"text-align: center;\">\\(\\begin{bmatrix}<br \/>\n3 &#038; -1 &#038; 0\\\\<br \/>\n2 &#038; 5 &#038; 3<br \/>\n\\end{bmatrix}<br \/>\n\\)<\/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\/\"><\/p>\n<p>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":100672,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-61508","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\/61508","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=61508"}],"version-history":[{"count":5,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/61508\/revisions"}],"predecessor-version":[{"id":292823,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/61508\/revisions\/292823"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/100672"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=61508"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}