Question:
If $A$ is a square matrix such that $A(\operatorname{adj} A)=\left[\begin{array}{lll}5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5\end{array}\right]$, then write the value of $|\operatorname{adj} A|$.
Solution:
Given : $A(\operatorname{adj} A)=\left[\begin{array}{lll}5 & 0 & 0\end{array}\right.$
$\begin{array}{lll}0 & 5 & 0\end{array}$
$\left.\begin{array}{lll}0 & 0 & 5\end{array}\right]$
$\Rightarrow|A| I_{n}=5\left[\begin{array}{lll}1 & 0 & 0\end{array}\right.$
$\begin{array}{lll}0 & 1 & 0\end{array}$
$\left.\begin{array}{lll}0 & 0 & 1\end{array}\right]$
$\Rightarrow|A|=5$
Now, $|\operatorname{adj} A|=|A|^{n-1}=5^{3-1}=25$
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