Question:
Let $A$ be a square matrix of order 3 and $B=|A| A^{-1}$. If $|\mathrm{A}|=-5$, then $|\mathrm{B}|=$_________
Solution:
Given:
$A$ is a square matrix of order 3
$B=|A| A^{-1}$
$|A|=-5$
Now,
$B=|A| A^{-1}$
$\Rightarrow|B|=|| A\left|A^{-1}\right|$
$\Rightarrow|B|=|A|^{3}\left|A^{-1}\right| \quad(\because$ order of $A$ is 3$)$
$\Rightarrow|B|=|A|^{3} \frac{1}{|A|} \quad\left(\because\left|A^{-1}\right|=\frac{1}{|A|}\right)$
$\Rightarrow|B|=|A|^{2}$
$\Rightarrow|B|=(-5)^{2} \quad(\because|A|=-5)$
$\Rightarrow|B|=25$
Hence, $|B|=\underline{25}$.