Question:
If $A$ is a square matrix of order 2 such that $A(\operatorname{adj} A)=\left[\begin{array}{cc}10 & 0 \\ 0 & 10\end{array}\right]$, then $|A|=$_)___________
Solution:
As we know that, $A(\operatorname{adj} A)=|A| I$.
But it is given that $A(\operatorname{adj} A)=\left[\begin{array}{cc}10 & 0 \\ 0 & 10\end{array}\right]$
$\Rightarrow A(\operatorname{adj} A)=10\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
$\Rightarrow A(\operatorname{adj} A)=10 I$
$\Rightarrow|A| I=10 I$
$\Rightarrow|A|=10$
Hence, $|A|=\underline{10}$.