Question

If $A$ is a square matrix of order $n$ such that $\left|adj\left(adjA\right)\right|={\left|A\right|}^9$, then the value of $n$ can be

• A. $4$
• B. $2$
• C. either $4$ or $2$
• D. None of these

Answer 1

Answer: (A)

$4$

We know that, in a square matrix of order $n$
$\left|adjA\right|={\left|A\right|}^{n-1}$
$\Rightarrow \left|adj\left(adjA\right)\right|={\left|adjA\right|}^{n-1}={\left|A\right|}^{{(n-1)}^2}$
$\Rightarrow n^2-2n-8=0$
$\Rightarrow n=4,-2$