Question

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

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

Answer 1

The correct option is A $4$

Given, $|adj\left(adjA\right)|=|{A|}^9$

We know that in a square matrix of order n,

$|adjA|=|A{|}^{n-1}$

$\Rightarrow |adj\left(adjA\right)|=|adjA{|}^{n-1}=|A{|}^{(n-1{)}^2}$

$\therefore |A{|}^{(n-1{)}^2}=|{A|}^9$
$\Rightarrow (n-1{)}^2=9$
$\Rightarrow n-1=3$
$\Rightarrow n=4$