Example D33M Determinant of a $3\times 3$ matrix

Suppose that we have the $3\times 3$ matrix \begin{equation*} A= \begin{bmatrix} 3 & 2 & -1\\ 4 & 1 & 6\\ -3 & -1 & 2 \end{bmatrix} \end{equation*}

Then \begin{align*} \detname{A}=\detbars{A} &=\begin{vmatrix} 3 & 2 & -1\\ 4 & 1 & 6\\ -3 & -1 & 2 \end{vmatrix}\\ &= 3 \begin{vmatrix} 1 & 6\\ -1 & 2 \end{vmatrix} -2 \begin{vmatrix} 4 & 6\\ -3 & 2 \end{vmatrix} +(-1) \begin{vmatrix} 4 & 1\\ -3 & -1 \end{vmatrix}\\ &= 3\left( 1\begin{vmatrix} 2\\ \end{vmatrix} -6\begin{vmatrix} -1 \end{vmatrix}\right) -2\left( 4\begin{vmatrix} 2 \end{vmatrix} -6\begin{vmatrix} -3 \end{vmatrix}\right) -\left( 4\begin{vmatrix} -1 \end{vmatrix} -1\begin{vmatrix} -3 \end{vmatrix}\right)\\ &= 3\left(1(2)-6(-1)\right) -2\left(4(2)-6(-3)\right) -\left(4(-1)-1(-3)\right)\\ &=24-52+1\\ &=-27 \end{align*}