Question:
If $A=\left[\begin{array}{ll}2 & 5 \\ 2 & 1\end{array}\right]$ and $B=\left[\begin{array}{cc}4 & -3 \\ 2 & 5\end{array}\right]$, verify that $|A B|=|A||B|$.
Solution:
Consider LHS
$A B=\left[\begin{array}{ll}2 & 5 \\ 2 & 1\end{array}\right]\left[\begin{array}{cc}4 & -3 \\ 2 & 5\end{array}\right]$
$=\left[\begin{array}{cc}8+10 & -6+25 \\ 8+2 & -6+5\end{array}\right]=\left[\begin{array}{cc}18 & 19 \\ 10 & -1\end{array}\right]$
$|A B|=-18-190=-208$
Consider RHS
$|A|=2-10=-8$
$|B|=20-(-6)=26$
$|A||B|=-8 \times 26=-208$
$\therefore$ LHS $=$ RHS