Question

Find the adjoint of the matrix $A=\left[\begin{array}{cc}1& 2\\ 3& -5\end{array}\right]$ and verify the result $A\left(adjA\right)=\left(adjA\right)A=|A|\cdot I$

Answer 1

$|A|=-11$
$adjA=\left[\begin{array}{cc}-5& -2\\ -3& 1\end{array}\right]$
$A\left(adjA\right)=\left[\begin{array}{cc}-11& 0\\ 0& -11\end{array}\right]$
$\left(adjA\right)A=\left[\begin{array}{cc}-11& 0\\ 0& -11\end{array}\right]$
$|A|I=\left[\begin{array}{cc}-11& 0\\ 0& -11\end{array}\right]$
$A\left(adjA\right)=\left(AdjA\right)A=|A|I$