Question:
If $A$ is a square matrix with $|A|=4$ then find the value of $|A .(a d j A)|$.
Solution:
Given:
$A$ is a square matrix
$|A|=4$
Now,
$A(\operatorname{adj} A)=|A| I$
$\Rightarrow|A(\operatorname{adj} A)|=|| A|I|$
$\Rightarrow|A(\operatorname{adj} A)|=|A|^{n}|I|$, where $n$ is the order of $I$
$\Rightarrow|A(\operatorname{adj} A)|=|A|^{n} \times 1$
$\Rightarrow|A(\operatorname{adj} A)|=4^{n}$ $(\because|A|=4)$
Hence, $|A(\operatorname{adj} A)|=\underline{4^{n}}$.
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