Question

For any $2\times 2$ matrix $A$, if $A\left(adjA\right)=\left[\begin{array}{cc}10& 0\\ 0& 10\end{array}\right]$, then $|A|$ i.e. $detA$ equals

Answer 1

$A\left(adj\right(A\left)\right)=|A|I_n$

$A\left(adjA\right)=|A|I_n=\left[\begin{array}{cc}10& 0\\ 0& 10\end{array}\right]$, then $|A|=10I_n$

$\Rightarrow det\left(A\right)=|A|=10$