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      <text>7.3.2.7 Evaluating general determinants example</text>
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      <text><![CDATA[<p>In the case of \(\Delta=\begin{vmatrix} 4 & 1 & 1 \\ 1 & 2 & 3 \\ 3 & 1 & 2 \end{vmatrix}\) we have \(a_{11}=4, a_{12}=1, a_{13}=1,\) and \begin{align*} A_{11} & = + \begin{vmatrix} 2 & 3 \\ 1 & 2 \end{vmatrix} = 4 - 3 = 1, \\ A_{12} & = - \begin{vmatrix} 1 & 3 \\ 3 & 2 \end{vmatrix} = - ( 2 - 9 ) = 7, \\ A_{13} & = + \begin{vmatrix} 1 & 2 \\ 3 & 1 \end{vmatrix} = 1 - 6 = - 5.\end{align*}</p> 

<p>Hence, \(\Delta=4\times 1 + 1\times 7 + 1\times -5 = 6\).</p>

<p>Alternatively, choosing to expand along the second row:
\begin{align*}\Delta & = a_{21} A_{21} + a_{22} A_{22} + a_{23} A_{23} \\ & = 1 \left( - \begin{vmatrix} 1 & 1 \\ 1 & 2 \end{vmatrix} \right) + 2 \left( \begin{vmatrix} 4 & 1 \\ 3 & 2 \end{vmatrix} \right) + 3 \left( - \begin{vmatrix} 4 & 1 \\ 3 & 1 \end{vmatrix} \right) \\ & = 6\end{align*}
as before.</p>]]></text>
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