Let A be a square matrix of order 3 and B

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, $|B|=\underline{25}$.

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