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  <question type="description">
    <name>
      <text>7.3.2.3 Introducing cofactors</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<h4> Cofactors </h4>
<p>A cofactor is a minor with a sign attached. The appropriate sign comes from the pattern of signs appropriate to a 
\(3\times 3\) array: \[\begin{bmatrix}+&amp;-&amp;+\\ -&amp;+&amp;-\\ +&amp;-&amp;+\end{bmatrix}\]
(i.e. positive signs on the leading diagonal and the signs "alternate" everywhere else.)</p>

<p>Each entry of a matrix \(A\) has a cofactor associated with it. The cofactor of entry \(a_{11}\)
is denoted by \(A_{11}\), that of \(a_{23}\)  by \(A_{23}\) and so on.</p>

<p>To obtain the cofactor of an element of a \(3\times 3\)  matrix we simply multiply the minor of that element by the corresponding sign from the \(3\times 3\) array of signs.</p>

<p>Hence, for the matrix \(A = \begin{bmatrix} 4 &amp; 1 &amp; 1 \\ 1 &amp; 2 &amp; 3 \\ 3 &amp; 1 &amp; 2 \end{bmatrix}\), the cofactor corresponding to \(a_{23}\) is
\[A_{23} = - \begin{vmatrix} 4 &amp; 1 \\ 3 &amp; 1 \end{vmatrix} = -1\] 
and the cofactor corresponding to \(a_{31}\) is
\[A_{31} = + \begin{vmatrix} 1 &amp; 1 \\ 2 &amp; 3 \end{vmatrix}=1.\]</p>]]></text>
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