If A is a square matrix with |A| = 4 then find the value

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, $|A(\operatorname{adj} A)|=\underline{4^{n}}$.