Question

Let $A$ be a square matrix such that $A\left(adjA\right)=\left[\begin{array}{ccc}4& 0& 0\\ 0& 4& 0\\ 0& 0& 4\end{array}\right].$ Then the value of $\frac{|adj\left(adjA\right)|}{|adjA|}$ is

Answer 1

$A\left(adjA\right)=|A|\cdot I_3=\left[\begin{array}{ccc}4& 0& 0\\ 0& 4& 0\\ 0& 0& 4\end{array}\right]$
$\Rightarrow |A|=4$

$|adj\left(adjA\right)|=|A{|}^{{(n-1)}^2}=4^4=256$
$|adjA|=|A{|}^{n-1}=4^2=16$
Now, $\frac{|adj\left(adjA\right)|}{|adjA|}=\frac{256}{16}=16$