Question

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

Answer 1

Given $A=\left[\begin{array}{cc}1& 2\\ 3& -5\end{array}\right]$ , so we get $\text{adj}A=\left[\begin{array}{cc}-5& -2\\ -3& 1\end{array}\right]$

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

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

$|A|=-11$

Therefore $A\left(\text{adj}A\right)=\left(\text{adj}A\right)A=|A|I$