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<h1 align="center">Inverse of a Matrix <br />
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using Elementary Row Operations</h1>
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<p class="center"><i>Also called the Gauss-Jordan method.</i></p>
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<p>This is a fun way to find the Inverse of a Matrix: </p>
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<div class="center simple">
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<div class="boxa"> Play around with the rows
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(adding, multiplying or swapping)
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until we make Matrix <b>A</b> into the Identity Matrix <b>I</b> </div>
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<div class="boxa"> <img src="images/matrix-gauss-jordan1.svg" alt="matrix A | I becomes I | A inverse" /> </div>
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<div class="boxa"> And by ALSO doing the changes to an Identity Matrix it magically turns into the Inverse! </div>
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</div>
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<p>The <b>"Elementary Row Operations"</b> are simple things like adding rows, multiplying and swapping ... but let's see with an example:</p>
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<h2>Example: find the Inverse of "A":</h2>
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<p align="center"><img src="images/matrix-gauss-jordan3.gif" width="173" height="73" alt="matrix A" /></p>
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<p>We start with the matrix <span class="larger">A</span>, and write it down with an Identity Matrix <span class="larger">I</span> next to it:</p>
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<p align="center"><img src="images/matrix-gauss-jordan6.gif" width="243" height="110" alt="matrix A augmented" /><br />
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(This is called the "Augmented Matrix")
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</p>
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<div class="center80">
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<h3>Identity Matrix</h3>
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<p>The "Identity Matrix" is the matrix equivalent of the number "1":</p>
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<p align="center"><img src="images/matrix-identity.gif" alt="Identity Matrix" width="186" height="95" /><br />
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<span class="large">A 3x3 Identity Matrix</span><br />
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</p>
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<ul>
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<li>It is "square" (has same number of rows as columns),</li>
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<li>It has <b>1</b>s on the diagonal and <b>0</b>s everywhere else. </li>
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<li>It's symbol is the capital letter <b>I</b>. </li>
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</ul>
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</div>
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<p> </p>
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<p>Now we do our best to turn "A" (the Matrix on the left) into an Identity Matrix. The goal is to make Matrix A have <b>1</b>s on the diagonal and <b>0</b>s elsewhere (an Identity Matrix) ... and the right hand side comes along for the ride, with every operation being done on it as well.</p>
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<p>But we can only do these <b>"Elementary Row Operations"</b>:</p>
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<ul>
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<li><b>swap</b> rows</li>
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<li><b>multiply</b> or divide each element in a a row by a constant</li>
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<li>replace a row by <b>adding</b> or subtracting a multiple of another row to it</li>
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</ul>
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<p>And we must do it to the <b>whole row</b>, like this: </p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/matrix-gauss-jordan2.svg" alt="matrix row steps" /></p>
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<div>
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<p> </p>
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<p>Start with <b>A</b> next to <b>I</b></p>
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<p> </p>
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<p>Add row 2 to row 1,</p>
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<p> </p>
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<p>then divide row 1 by 5,</p>
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<p> </p>
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<p>Then take 2 times the first row, and subtract it from the second row,</p>
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<p> </p>
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<p>Multiply second row by -1/2,</p>
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<p> </p>
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<p>Now swap the second and third row,</p>
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<p> </p>
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<p>Last, subtract the third row from the second row,</p>
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<p>And we are done!</p>
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</div>
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<p> </p>
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<p align="left" class="larger">And matrix <b>A</b> has been made into an Identity Matrix ...</p>
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<p align="center" class="larger">... and at the same time an Identity Matrix got made into <b>A<sup>-1</sup></b></p>
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<p align="center"><img src="images/matrix-gauss-jordan4.gif" width="189" height="72" alt="matrix A inverse" /></p>
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<p align="center">DONE! Like magic, and just as fun as solving any puzzle.</p>
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<p align="left"><b>And note: there is no "right way" to do this, just keep playing around until we succeed!</b></p>
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<p align="left">(Compare this answer with the one we got on <a href="matrix-inverse-minors-cofactors-adjugate.html">Inverse of a Matrix using Minors, Cofactors and Adjugate</a>. Is it the same? Which method do you prefer?)</p>
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<h2 align="left">Larger Matrices</h2>
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<p align="left">We can do this with larger matrices, for example, try this 4x4 matrix:</p>
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<p align="center"><img src="images/matrix-gauss-jordan8.gif" width="175" height="72" alt="matrix B" /></p>
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<p>Start Like this:</p>
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<p align="center"><img src="images/matrix-gauss-jordan9.gif" width="243" height="70" alt="matrix B augmented" /></p>
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<p>See if you can do it yourself (I would begin by dividing the first row by 4, but you do it your way).</p>
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<p>You can check your answer using the <a href="matrix-calculator.html">Matrix Calculator</a> (use the "inv(A)" button).<br />
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</p>
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<h2>Why it Works</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/matrix-gauss-jordan5.svg" alt="8|1 becomes 1|(1/8)" /></p>
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<p>I like to think of it this way: </p>
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<ul>
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<li>when we turn "8" into "1" by dividing by 8, </li>
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<li>and do the same thing to "1", it turns into "1/8"</li>
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</ul>
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<p>And "1/8" is the (multiplicative)<b> inverse of 8</b></p>
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<div style="clear:both"></div>
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<p> </p>
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<p>Or, more technically:</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/matrix-gauss-jordan7.svg" alt="matrix A | I becomes I | A inverse" /></p>
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<p>The <b>total effect of all the row operations</b> is the same as <b>multiplying by <span class="larger">A<sup>-1</sup></span></b></p>
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<p>So <b><span class="larger">A</span></b> becomes <span class="larger">I</span> (because <b><span class="larger">A<sup>-1</sup></span></b><b><span class="larger">A</span></b> = <b><span class="larger">I</span></b>)<br />
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And <span class="larger">I</span> becomes <b><span class="larger">A<sup>-1</sup></span></b> (because <b><span class="larger">A<sup>-1</sup></span></b><b><span class="larger">I</span></b> = <b><span class="larger">A<sup>-1</sup></span></b>)</p>
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<div style="clear:both"></div>
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<p> </p>
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<a href="matrix-multiplying.html">Multiplying Matrices</a>
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<a href="matrix-determinant.html">Determinant of a Matrix</a>
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<a href="matrix-inverse-row-operations-gauss-jordan.html">Inverse of a Matrix
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using Elementary Row Operations</a>
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<a href="index.html">Algebra Index</a> </div>
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