Question
If $A=\left[\begin{array}{ll}2 & 1 \\ 0 & 3\end{array}\right], B=\left[\begin{array}{cc}1 & 2 \\ 3 & -2\end{array}\right]$, verify that $|A B|=|A||| B \mid$.
✓
Answer
$\begin{aligned} \mathrm{AB} & =\left[\begin{array}{ll}2 & 1 \\ 0 & 3\end{array}\right]\left[\begin{array}{cc}1 & 2 \\ 3 & -2\end{array}\right] \\ & =\left[\begin{array}{cc}2+3 & 4-2 \\ 0+9 & 0-6\end{array}\right]=\left[\begin{array}{cc}5 & 2 \\ 9 & -6\end{array}\right] \\ |\mathrm{AB}| & =\left[\begin{array}{cc}5 & 2 \\ 9 & -6\end{array}\right] \\ |\mathrm{AB}| & =5(-6)-9(2)=-30-18=-48\end{aligned}$
$|A|=\left|\begin{array}{ll}2 & 1 \\ 0 & 3\end{array}\right|=2(3)-0(1)=6+0=6$
$\begin{aligned} & |B|=\left|\begin{array}{cc}1 & 2 \\ 3 & -2\end{array}\right|=1(-2)-2(3)=-2-6=-8 \\ & |A| \cdot|B|=6(-8)=-48=|A B| \\ & |A B|=|A| \cdot|B|\end{aligned}$
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