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<div class="title">DET Determinant of a Matrix </div> </div>
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<div class="textblock"><p>Section: <a class="el" href="sec_array.html">Array Generation and Manipulations</a> </p>
<h1><a class="anchor" id="Usage"></a>
Usage</h1>
<p>Calculates the determinant of a matrix. Note that for all but very small problems, the determinant is not particularly useful. The condition number <code>cond</code> gives a more reasonable estimate as to the suitability of a matrix for inversion than comparing <code>det(A)</code> to zero. In any case, the syntax for its use is </p>
<pre class="fragment"> y = det(A)
</pre><p> where A is a square matrix. </p>
<h1><a class="anchor" id="Function"></a>
Internals</h1>
<p>The determinant is calculated via the <code>LU</code> decomposition. Note that the determinant of a product of matrices is the product of the determinants. Then, we have that </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ L U = P A \]" src="form_1.png"/>
</p>
<p> where <code>L</code> is lower triangular with 1s on the main diagonal, <code>U</code> is upper triangular, and <code>P</code> is a row-permutation matrix. Taking the determinant of both sides yields </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ |L U| = |L| |U| = |U| = |P A| = |P| |A| \]" src="form_2.png"/>
</p>
<p> where we have used the fact that the determinant of <code>L</code> is 1. The determinant of <code>P</code> (which is a row exchange matrix) is either 1 or -1. </p>
<h1><a class="anchor" id="Example"></a>
Example</h1>
<p>Here we assemble a random matrix and compute its determinant</p>
<pre class="fragment">--> A = rand(5);
--> det(A)
ans =
-0.0489
</pre><p>Then, we exchange two rows of <code>A</code> to demonstrate how the determinant changes sign (but the magnitude is the same)</p>
<pre class="fragment">--> B = A([2,1,3,4,5],:);
--> det(B)
ans =
0.0489
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