Question
If $A=\left[\begin{array}{cc}1 & 2 \\ 3 & 2 \\ -1 & 0\end{array}\right]$ and $B=\left[\begin{array}{ccc}1 & 3 & 2 \\ 4 & -1 & -3\end{array}\right]$, show that $A B$ is singular.
$=\left[\begin{array}{ccc}1+8 & 3-2 & 2-6 \\ 3+8 & 9-2 & 6-6 \\ -1+0 & -3+0 & -2+0\end{array}\right]$
$=\left[\begin{array}{ccc}9 & 1 & -4 \\ 11 & 7 & 0 \\ -1 & -3 & -2\end{array}\right]$
$|A B|=\left|\begin{array}{ccc}9 & 1 & -4 \\ 11 & 7 & 0 \\ -1 & -3 & -2\end{array}\right|$
$\begin{aligned} & =9(-14+0)-1(-22+0)-4(-33+7) \\ & =-126+22+104\end{aligned}$
$=0$
$\mathrm{AB}$ is a singular matrix.
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