6.4 - The Determinant of a Square Matrix

6.4 - The Determinant of a Square Matrix - <?xml...

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<?xml version="1.0" encoding="iso-8859-1"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title>6.4 - The Determinant of a Square Matrix</title> <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" /> <link href=". ./m116.css" rel="stylesheet" type="text/css" /> <link href="matrix.css" rel="stylesheet" type="text/css" /> </head> <body> <h1>6.4 - The Determinant of a Square Matrix</h1> <p>A determinant is a real number associated with every square matrix. I have yet to find a good English definition for what a determinant is. Everything I can find either defines it in terms of a mathematical formula or suggests some of the uses of it. There's even a definition of determinant that defines it in terms of itself. </p> <p>The determinant of a square matrix A is denoted by "det A" or | A |. Now, that last one looks like the absolute value of A, but you will have to apply context. If the vertical lines are around a matrix, it means determinant.</p> <p>The line below shows the two ways to write a determinant.</p> <table border="0" cellspacing="0" cellpadding="1"> <tr> <td class="detl">3</td> <td class="detr">1</td> <td rowspan="2" class="downhalf">=</td> <td rowspan="2" class="downhalf">det</td> <td class="matlt"> </td> <td class="lhs">3</td> <td class="lhs">1</td> <td class="matrt"> </td> </tr> <tr> <td class="detl">5</td> <td class="detr">2</td> <td class="matlb"> </td> <td class="lhs">5</td> <td class="lhs">2</td> <td class="matrb"> </td> </tr> </table> <h2>Determinant of a 2×2 Matrix</h2> <p>The determinant of a 2&times;2 matrix is found much like a <a href="pivot.html">pivot</a> operation. It is the product of the elements on the main diagonal minus the product of the elements off the main diagonal.</p> <table border="0" cellspacing="0" cellpadding="1"> <tr> <td class="detl">a</td> <td class="detr">b</td> <td rowspan="2" class="downhalf">= ad - bc</td> </tr> <tr>
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<td class="detl">c</td> <td class="detr">d</td> </tr> </table> <h2>Properties of Determinants</h2> <ul> <li>The determinant is a real number, it is not a matrix.</li> <li>The determinant can be a negative number.</li> <li>It is not associated with absolute value at all except that they both use vertical lines.</li> <li>The determinant only exists for square matrices (2×2, 3×3, ... n×n). The determinant of a 1×1 matrix is that single value in the determinant.</li> <li>The inverse of a matrix will exist only if the determinant is not zero.</li> </ul> <h2>Expansion using Minors and Cofactors</h2> <p>The definition of determinant that we have so far is only for a 2&times;2 matrix. There is a shortcut for a 3&times;3 matrix, but I firmly believe you should learn the way that will work for all sizes, not just a special case for a 3×3 matrix. </p>
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