e39465ad2f
new file: Files/flashplayer_32_sa.exe
new file: favicon.ico
new file: globe.gif
new file: imgs/download.png
new file: imgs/zuck.jpg
new file: index.html
new file: other.ico
new file: script.js
new file: site.webmanifest
new file: sitemap.html
new file: styles/backround.css
new file: styles/border.css
new file: styles/fonts/Titillium_Web/OFL.txt
new file: styles/fonts/Titillium_Web/TitilliumWeb-Black.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Bold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-BoldItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLight.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Italic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Light.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-LightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Regular.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBoldItalic.ttf
new file: styles/fonts/webfontkit-20221027-163353/generator_config.txt
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-demo.html
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff2
new file: styles/fonts/webfontkit-20221027-165950/generator_config.txt
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-demo.html
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff2
new file: styles/style.css
new file: tools/2048/.gitignore
new file: tools/2048/.jshintrc
new file: tools/2048/CONTRIBUTING.md
new file: tools/2048/LICENSE.txt
new file: tools/2048/README.md
new file: tools/2048/Rakefile
new file: tools/2048/favicon.ico
new file: tools/2048/index.html
new file: tools/2048/js/animframe_polyfill.js
new file: tools/2048/js/application.js
new file: tools/2048/js/bind_polyfill.js
new file: tools/2048/js/classlist_polyfill.js
new file: tools/2048/js/game_manager.js
new file: tools/2048/js/grid.js
new file: tools/2048/js/html_actuator.js
new file: tools/2048/js/keyboard_input_manager.js
new file: tools/2048/js/local_storage_manager.js
new file: tools/2048/js/tile.js
new file: tools/2048/meta/apple-touch-icon.png
new file: tools/webretro/cores/neocd_libretro.js
new file: tools/webretro/cores/neocd_libretro.wasm
new file: tools/webretro/cores/nestopia_libretro.js
new file: tools/webretro/cores/nestopia_libretro.wasm
new file: tools/webretro/cores/o2em_libretro.js
new file: tools/webretro/cores/o2em_libretro.wasm
new file: tools/webretro/cores/opera_libretro.js
new file: tools/webretro/cores/opera_libretro.wasm
269 lines
10 KiB
HTML
269 lines
10 KiB
HTML
<!DOCTYPE html>
|
||
<html lang="en"><!-- #BeginTemplate "/Templates/Advanced.dwt" --><!-- DW6 -->
|
||
|
||
<!-- Mirrored from www.mathsisfun.com/algebra/matrix-inverse-minors-cofactors-adjugate.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:59:46 GMT -->
|
||
<head>
|
||
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
|
||
|
||
|
||
<!-- #BeginEditable "doctitle" -->
|
||
<title>Inverse of a Matrix using Minors, Cofactors and Adjugate</title>
|
||
|
||
<meta name="Description" content="Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.">
|
||
<style type="text/css">
|
||
.style1 {color: #9933FF}
|
||
.style2 {color: #990000}
|
||
|
||
.mat {
|
||
font: 20px Verdana, Arial, Trebuchet MS, Tahoma, Geneva, Verdana, sans-serif;
|
||
color: #f30;
|
||
|
||
background-image: linear-gradient(#07c, #07c),
|
||
linear-gradient(#07c, #07c),
|
||
linear-gradient(#07c, #07c),
|
||
linear-gradient(#07c, #07c);
|
||
|
||
background-repeat: no-repeat;
|
||
background-size: 7px 2px;
|
||
background-position: top left, top right, bottom left, bottom right;
|
||
|
||
border: solid #07c;
|
||
|
||
border-width: 0 2px;
|
||
display: inline-block;
|
||
vertical-align: middle;
|
||
|
||
padding: 2px 9px 3px 9px;
|
||
|
||
border-radius:3px;
|
||
|
||
}
|
||
|
||
|
||
.cols1, .cols2, .cols3, .cols4 {
|
||
display: inline-grid;
|
||
grid-template-columns: max-content;
|
||
align-content: space-evenly;
|
||
grid-gap: 4px 14px;
|
||
text-align: center;
|
||
vertical-align: middle;
|
||
}
|
||
|
||
.cols1 { grid-template-columns: max-content; }
|
||
.cols2 { grid-template-columns: repeat(2, max-content); }
|
||
.cols3 { grid-template-columns: repeat(3, max-content); }
|
||
.cols4 { grid-template-columns: repeat(4, max-content); }
|
||
|
||
.txt {
|
||
display: inline-block;
|
||
vertical-align: middle;
|
||
padding: 3px 3px;
|
||
font: 1.5rem Verdana, Geneva, sans-serif;
|
||
color: orange;
|
||
text-align: center;
|
||
min-width: 25px;
|
||
}
|
||
.txt .intbl {font-size: 90%; }
|
||
|
||
.inv { color:#07c; display:inline-block; transform: translateX(-0.1em) translateY(-1.1em); font-size: 90%; }
|
||
|
||
</style>
|
||
|
||
|
||
|
||
|
||
|
||
<!-- #EndEditable -->
|
||
<meta name="keywords" content="math, maths, mathematics, school, homework, education">
|
||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
|
||
<meta name="HandheldFriendly" content="true">
|
||
<meta name="referrer" content="always">
|
||
<link rel="preload" href="../images/style/font-champ-bold.ttf" as="font" type="font/ttf" crossorigin="">
|
||
<link rel="preload" href="../style4.css" as="style">
|
||
<link rel="preload" href="../main4.js" as="script">
|
||
<link rel="stylesheet" href="../style4.css">
|
||
<script src="../main4.js" defer="defer"></script>
|
||
<!-- Global site tag (gtag.js) - Google Analytics -->
|
||
<script async="" src="https://www.googletagmanager.com/gtag/js?id=UA-29771508-1"></script>
|
||
<script>
|
||
window.dataLayer = window.dataLayer || [];
|
||
function gtag(){dataLayer.push(arguments);}
|
||
gtag('js', new Date());
|
||
gtag('config', 'UA-29771508-1');
|
||
</script>
|
||
</head>
|
||
|
||
<body id="bodybg" class="adv">
|
||
|
||
<div id="stt"></div>
|
||
<div id="adTop"></div>
|
||
<header>
|
||
<div id="hdr"></div>
|
||
<div id="tran"></div>
|
||
<div id="adHide"></div>
|
||
<div id="cookOK"></div>
|
||
</header>
|
||
|
||
<div class="mid">
|
||
|
||
<nav>
|
||
<div id="menuWide" class="menu"></div>
|
||
<div id="logo"><a href="../index.html"><img src="../images/style/logo-adv.svg" alt="Math is Fun Advanced"></a></div>
|
||
|
||
<div id="search" role="search"></div>
|
||
<div id="linkto"></div>
|
||
|
||
<div id="menuSlim" class="menu"></div>
|
||
<div id="menuTiny" class="menu"></div>
|
||
</nav>
|
||
|
||
<div id="extra"></div>
|
||
|
||
<article id="content" role="main">
|
||
|
||
<!-- #BeginEditable "Body" -->
|
||
|
||
|
||
<h1 class="center">Inverse of a Matrix<br>
|
||
using Minors, Cofactors and Adjugate</h1>
|
||
|
||
<p class="center"><i>Note: also check out <a href="matrix-inverse-row-operations-gauss-jordan.html">Matrix Inverse by Row Operations</a> and the <a href="matrix-calculator.html">Matrix Calculator</a></i></p>
|
||
<p> </p>
|
||
<p class="larger">We can calculate the <a href="matrix-inverse.html">Inverse of a Matrix</a> by:</p>
|
||
<ul>
|
||
<li>Step 1: calculating the Matrix of Minors,</li>
|
||
<li>Step 2: then turn that into the Matrix of Cofactors,</li>
|
||
<li>Step 3: then the Adjugate, and</li>
|
||
<li>Step 4: multiply that by 1/Determinant.</li>
|
||
</ul>
|
||
<p class="larger">But it is best explained by working through an example!</p>
|
||
|
||
|
||
<h2>Example: find the Inverse of A:</h2>
|
||
|
||
|
||
<div style="text-align: center;">
|
||
<div class="txt">A =</div>
|
||
<div class="mat">
|
||
<div class="cols3">
|
||
<span>3</span><span>0</span><span>2</span>
|
||
<span>2</span><span>0</span><span>-2</span>
|
||
<span>0</span><span>1</span><span>1</span>
|
||
</div>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<p>It needs 4 steps. It is all simple arithmetic but there is a lot of it, so try not to make a mistake!</p>
|
||
|
||
|
||
<h2>Step 1: Matrix of Minors</h2>
|
||
|
||
<p>The first step is to create a "Matrix of Minors". This step has the most calculations.</p>
|
||
<p>For each element of the matrix:</p>
|
||
<ul>
|
||
<li>ignore the values on the current row and column</li>
|
||
<li><a href="matrix-determinant.html">calculate the determinant</a> of the remaining values</li>
|
||
</ul>
|
||
<p>Put those determinants into a matrix (the "Matrix of Minors")</p>
|
||
<div class="center80">
|
||
|
||
<h3>Determinant</h3>
|
||
<p>For a 2×2 matrix (2 rows and 2 columns) the determinant is easy: <b>ad-bc</b></p>
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td>
|
||
<p class="indent50px">Think of a cross:</p>
|
||
<ul>
|
||
<li class="indent50px">Blue means positive (+ad),</li>
|
||
<li class="indent50px">Red means negative (-bc)</li>
|
||
</ul></td>
|
||
<td> </td>
|
||
<td class="indent50px"><img src="images/matrix-2x2-det-c.gif" alt="A Matrix" height="69" width="95"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>(It gets harder for a 3×3 matrix, etc)</p>
|
||
</div>
|
||
|
||
<h3>The Calculations</h3>
|
||
<p>Here are the first two, and last two, calculations of the "<b>Matrix of Minors</b>" (notice how I ignore the values in the current row and columns, and calculate the determinant using the remaining values):</p>
|
||
<p class="center"><img src="images/matrix-minors1.gif" alt="matrix of minors calculation steps" height="353" width="273"></p>
|
||
<p>And here is the calculation for the whole matrix:</p>
|
||
<p class="center"><img src="images/matrix-minors2.svg" alt="matrix minors result" height="91" width="559"></p>
|
||
|
||
|
||
<h2>Step 2: Matrix of Cofactors</h2>
|
||
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/matrix-checkerboard.svg" alt="checkerboard of plus and minus" height="120" width="120"></p>
|
||
<p>This is easy! Just apply a "checkerboard" of minuses to the "Matrix of Minors". In other words, we need to change the sign of alternate cells, like this:</p>
|
||
<p class="center"><img src="images/matrix-cofactors.svg" alt="matrix of cofactors" height="91" width="468"></p>
|
||
|
||
|
||
<h2>Step 3: Adjugate (also called Adjoint)</h2>
|
||
|
||
<p>Now "Transpose" all elements of the previous matrix... in other words swap their positions over the diagonal (the diagonal stays the same):</p>
|
||
<p class="center"><img src="images/matrix-adjugate.gif" alt="matrix adjugate" height="70" width="123"></p>
|
||
|
||
|
||
<h2>Step 4: Multiply by 1/Determinant</h2>
|
||
|
||
<p>Now <a href="matrix-determinant.html">find the determinant</a> of the original matrix. This isn't too hard, because we already calculated the determinants of the smaller parts when we did "Matrix of Minors".</p>
|
||
<p class="center"><span class="larger"><img src="images/matrix-3x3-det.svg" alt="A Matrix" height="104" width="406"></span></p>
|
||
<p>Using:</p>
|
||
<p class="center">Elements of top row: 3, 0, 2<br>
|
||
Minors for top row: 2, 2, 2</p>
|
||
|
||
<p>We end up with this calculation:</p>
|
||
|
||
<p class="center larger">Determinant = 3×2 − 0×2 + 2×2 = <b>10</b></p>
|
||
<div class="info">
|
||
<p><b>Note:</b> a small simplification is to multiply by the cofactors (which already have the "+−+−" pattern), and then we just add each time:</p>
|
||
<p class="center larger">Determinant = 3×2 <b class="hilite">+</b> 0×(<b class="hilite">−</b>2) + 2×2 = <b>10</b></p>
|
||
|
||
</div><p><br></p>
|
||
<p class="center80">Your Turn: try this for <b>any other row or column</b>, you should also get 10.</p>
|
||
<p><br></p>
|
||
<p>Now we multiply the Adjugate by 1/Determinant to get:</p>
|
||
<p class="center"><img src="images/matrix-adjugate-inverse.svg" alt="matrix adjugate by 1/det gives inverse" height="95" width="383"></p>
|
||
<p class="center large">And we are done!</p>
|
||
<p>Compare this answer with the one we got on <a href="matrix-inverse-row-operations-gauss-jordan.html">Inverse of a Matrix
|
||
using Elementary Row Operations</a>. Is it the same? Which method do you prefer?</p>
|
||
|
||
|
||
<h2>Larger Matrices</h2>
|
||
|
||
<p>It is exactly the same steps for larger matrices (such as a 4×4, 5×5, etc), but wow! there is a lot of calculation involved.</p>
|
||
<p>For a 4×4 Matrix we have to calculate 16 3×3 determinants. So it is often easier to use computers (such as the <a href="matrix-calculator.html">Matrix Calculator</a>.)</p>
|
||
|
||
|
||
<h2>Conclusion</h2>
|
||
|
||
<ul class="larger">
|
||
<li>For each element, calculate the <b>determinant of the values not on the row or column</b>, to make the Matrix of Minors</li>
|
||
<li>Apply a <b>checkerboard</b> of minuses to make the Matrix of Cofactors</li>
|
||
<li><b>Transpose</b> to make the Adjugate</li>
|
||
<li>Multiply by <b>1/Determinant</b> to make the Inverse</li>
|
||
</ul>
|
||
<p> </p>
|
||
<div class="questions">2617, 2618, 8500, 8501, 8502, 8503, 8504, 8505, 8506, 8507</div>
|
||
|
||
<div class="related">
|
||
<a href="matrix-multiplying.html">Multiplying Matrices</a>
|
||
<a href="matrix-determinant.html">Determinant of a Matrix</a>
|
||
<a href="matrix-calculator.html">Matrix Calculator</a>
|
||
<a href="index.html">Algebra Index</a>
|
||
</div>
|
||
<!-- #EndEditable -->
|
||
|
||
</article>
|
||
|
||
<div id="adend" class="centerfull noprint"></div>
|
||
<footer id="footer" class="centerfull noprint"></footer>
|
||
<div id="copyrt">Copyright © 2022 Rod Pierce</div>
|
||
|
||
</div>
|
||
</body><!-- #EndTemplate -->
|
||
<!-- Mirrored from www.mathsisfun.com/algebra/matrix-inverse-minors-cofactors-adjugate.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:59:47 GMT -->
|
||
</html> |