Question

If for any $2\times 2$ square matrix A, $A\left(adjA\right)=\left[\begin{array}{cc}8& 0\\ 0& 8\end{array}\right]$, them write the value of det[A].

Answer 1

let A =$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

Adj(A) = $\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

Now $A\left(AdjA\right)=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\ast \left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}8& 0\\ 0& 8\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\ast \left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

$\Rightarrow ad-bc=8,-ab+ab=0,dc-dc=0,-bc+ad=8$

$det\left(A\right)=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc=8$