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<h1 class="center">Determinant of a Matrix</h1>
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<p>The determinant is a <b>special number</b> that can be calculated from a <a href="matrix-introduction.html">matrix</a>.</p>
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<p>The matrix has to be square (same number of rows and columns) like this one:</p>
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<div style="text-align: center; transform: scale(1.2);">
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<div class="mat">
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<div class="cols2">
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<span>3</span><span>8</span>
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<span>4</span><span>6</span>
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</div>
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</div>
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</div>
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<!-- [3,8~4,6] -->
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<p class="center"><span class="large">A Matrix</span><br>
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(This one has 2 Rows and 2 Columns)</p>
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<p>Let us calculate the determinant of that matrix:</p>
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<p class="center larger">3×6 − 8×4<br> = 18 − 32<br> = <b>−14</b></p>
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<p>Easy, hey? Here is another example:</p>
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<div class="example">
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<h3>Example:
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<div style="text-align: center;">
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<div class="txt">B =</div>
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<div class="mat">
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<div class="cols2">
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<span>1</span><span>2</span>
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<span>3</span><span>4</span>
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</div>
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</div>
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</div>
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<!-- A = [1,2~3,4] --></h3>
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<p>The <b>symbol</b> for determinant is two vertical lines either side like this:</p>
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<p class="center larger"><b>|B|</b> = 1×4 − 2×3<br>= 4 − 6<br> = <b>−2</b></p>
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<p>(Note: it is the same symbol as <a href="../numbers/absolute-value.html">absolute value</a>.)</p>
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</div>
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<h2>What is it for?</h2>
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<p>The determinant helps us find the <a href="matrix-inverse-minors-cofactors-adjugate.html">inverse of a matrix</a>, tells us things about the matrix that are useful in <a href="systems-linear-equations.html">systems of linear equations</a>, <a href="../calculus/index.html">calculus</a> and more.</p>
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<h2>Calculating the Determinant</h2>
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<p>First of all the matrix must be <b>square</b> (i.e. have the same number of rows as columns). Then it is just arithmetic. </p>
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<h2>For a 2×2 Matrix</h2>
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<p>For a <span class="number large">2×2</span> matrix (2 rows and 2 columns):</p>
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<div style="text-align: center;">
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<div class="txt">A =</div>
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<div class="mat">
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<div class="cols2">
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<span>a</span><span>b</span>
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<span>c</span><span>d</span>
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</div>
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</div>
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</div>
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<!-- A = [a,b~c,d] -->
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<p>The determinant is:</p>
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<p class="center"><span class="larger">|A| = ad − bc</span><span class="large"><br>
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</span><i>"The determinant of A equals a times d minus b times c"</i></p>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td>
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<p>It is easy to remember when you think of a cross:</p>
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<ul>
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<li><span class="style1">Blue</span> is positive (+ad),</li>
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<li><span class="style2">Red</span> is negative (−bc)</li>
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</ul></td>
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<td> </td>
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<td>
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<img src="images/matrix-2x2-det.svg" alt="a by d, b by c">
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</td>
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</tr>
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</tbody></table><br>
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<div class="example">
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<h3>Example: find the determinant of
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<div style="text-align: center;">
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<div class="txt">C =</div>
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<div class="mat">
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<div class="cols2">
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<span>4</span><span>6</span>
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<span>3</span><span>8</span>
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</div>
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</div>
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</div>
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<!-- B = [4,6~3,8] -->
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</h3>
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<p>
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Answer:
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</p>
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<div class="tbl">
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<div class="row"><span class="left"><span class="larger"><span class="large">|C|</span></span></span><span class="right"><span class="larger"><span class="large">= 4×8 − 6×3</span></span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="larger"><span class="large">= 32 − 18</span></span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="larger"><span class="large">= 14</span></span></span></div>
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</div>
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</div>
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<h2>For a 3×3 Matrix</h2>
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<p>For a <span class="number large">3×3</span> matrix (3 rows and 3 columns):</p>
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<div style="text-align: center;">
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<div class="txt">A =</div>
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<div class="mat">
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<div class="cols3">
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<span>a</span><span>b</span><span>c</span>
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<span>d</span><span>e</span><span>f</span>
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<span>g</span><span>h</span><span>i</span>
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</div>
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</div>
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</div>
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<!-- A = [a,b,c~d,e,f~g,h,i] -->
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<p>The determinant is:</p>
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<p class="center"><span class="larger">|A| = a(ei − fh) − b(di − fg) + c(dh − eg)</span><span class="large"><br>
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</span><i>"The determinant of A equals ... etc"</i></p>
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<p>It may look complicated, but<b> there is a pattern</b>:</p>
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<p class="center">
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<img src="images/matrix-3x3-det.svg" alt="multiply pattern"> </p>
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<p>To work out the determinant of a <b>3×3</b> matrix:</p>
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<ul>
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<li>Multiply <b>a</b> by the <b>determinant of the 2×2 matrix</b> that is<b> not in a</b>'s row or column.</li>
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<li>Likewise for <b>b</b>, and for <b>c</b></li>
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<li>Sum them up, but remember the minus in front of the <b>b</b></li>
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</ul>
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<p>As a formula <i>(remember the vertical bars</i> ||<i> mean "determinant of")</i>:</p>
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<p class="center"><img style="background-color: hsl(240,100%,92%);" src="images/matrix-3x3-det-latex.gif" alt="A Matrix" height="56" width="329"><br>
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<i>"The determinant of A equals a times the determinant of ... etc"</i></p>
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<div class="example">
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<h3>Example:
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<div style="text-align: center;">
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<div class="txt">D =</div>
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<div class="mat">
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<div class="cols3">
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<span>6</span><span>1</span><span>1</span>
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<span>4</span><span>−2</span><span>5</span>
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<span>2</span><span>8</span><span>7</span>
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</div>
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</div>
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</div>
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<!-- C = [6,1,1~4,-2,5~2,8,7] -->
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</h3>
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<div class="tbl">
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<div class="row"><span class="left"><span class="large"><b>|D|</b></span></span><span class="right"><span class="large">= 6×(−2×7 − 5×8) − 1×(4×7 − 5×2) + 1×(4×8 − (−2×2))</span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="large">= 6×(−54) − 1×(18) + 1×(36)</span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="large">= <b>−306</b></span></span></div>
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</div>
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</div>
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<h2>For 4×4 Matrices and Higher</h2>
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<p>The pattern continues for <span class="number">4×4</span> matrices:</p>
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<ul>
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<li><b>plus</b> <b>a</b> times the determinant of the matrix that is<b> not</b> in <b>a</b>'s row or column,</li>
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<li><b>minus b</b> times the determinant of the matrix that is<b> not</b> in <b>b</b>'s row or column,</li>
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<li><b>plus c</b> times the determinant of the matrix that is<b> not</b> in <b>c</b>'s row or column,</li>
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<li><b>minus d</b> times the determinant of the matrix that is<b> not</b> in <b>d</b>'s row or column,</li>
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</ul>
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<p class="center">
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<img src="images/matrix-4x4-det.svg" alt="multiply pattern"> </p><p></p>
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<p>As a formula:</p>
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<p class="center"><img style="background-color: hsl(240,100%,92%);" src="images/matrix-4x4-det-latex.gif" alt="4x4 determinant formula" height="79" width="561"></p>
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<p class="center">Notice the <b>+−+−</b> pattern (<span class="hilite">+</span>a... <span class="hilite">−</span>b... <span class="hilite">+</span>c... <span class="hilite">−</span>d...). This is important to remember.</p>
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<p> </p>
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<p>The pattern continues for <span class="number">5×5</span> matrices and higher. Usually best to use a <a href="matrix-calculator.html">Matrix Calculator</a> for those!</p>
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<p> </p>
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<h2>Not The Only Way</h2>
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<p>This method of calculation is called the "Laplace expansion" and I like it because the pattern is easy to remember. But there are other methods (just so you know).</p>
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<h2>Summary</h2>
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<ul>
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<div class="bigul">
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<li>For a <span class="number">2×2</span> matrix the determinant is<b> ad - bc</b></li>
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<li>For a <span class="number">3×3</span> matrix multiply <b>a</b> by the <b>determinant of the 2×2 matrix</b> that is<b> not</b> in <b>a</b>'s row or column, likewise for <b>b</b> and <b>c</b>, but remember that <b>b</b> has a negative sign!</li>
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<li>The pattern continues for larger matrices: multiply <b>a</b> by the <b>determinant of the matrix</b> that is<b> not</b> in <b>a</b>'s row or column, continue like this across the whole row, but remember the + − + − pattern.</li>
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</div>
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</ul>
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<p> </p>
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<div class="questions">718,2390,2391,2392,8477,719,2393,8478,8479,8480</div>
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<div class="related">
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<a href="matrix-introduction.html">Matrices</a>
|
||
<a href="matrix-calculator.html">Matrix Calculator</a>
|
||
<a href="index.html">Algebra Index</a>
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</div>
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