Question

Verify $A\left(adj\right)\left(A\right)-\left(adjA\right)A=|A|I_n|)in$

$\left[\begin{array}{cc}2& 3\\ -4& -6\end{array}\right]$

Answer 1

Let $A=\left[\begin{array}{cc}2& 3\\ -4& -6\end{array}\right],|A|=\left|\begin{array}{cc}2& 3\\ -4& -6\end{array}\right|=-12-(-12)=-12+12=0$
$\therefore |A|I=0\left|\begin{array}{cc}1& 0\\ 0& 1\end{array}\right|=\left|\begin{array}{cc}0& 0\\ 0& 0\end{array}\right|=0$
Cofactors A are $A_{11}=-6,A_{12}=4,A_{21}=-3,A_{22}=2$
$\therefore adj\left(A\right)={\left|\begin{array}{cc}-6& 4\\ -3& 2\end{array}\right|}^T=\left|\begin{array}{cc}-6& -3\\ 4& 2\end{array}\right|$
Now, $\left(adjA\right)A=\left|\begin{array}{cc}-6& -3\\ 4& 2\end{array}\right|\left|\begin{array}{cc}2& 3\\ -4& -6\end{array}\right|=\left|\begin{array}{cc}-12+12& -18+18\\ 8-8& 12-12\end{array}\right|=\left|\begin{array}{cc}0& 0\\ 0& 0\end{array}\right|=0$
Also, $A\left(adjA\right)=\left|\begin{array}{cc}2& 3\\ -4& -6\end{array}\right|\left|\begin{array}{cc}-6& -3\\ 4& 2\end{array}\right|=\left|\begin{array}{cc}-12+12& -6+6\\ 24-24& 12-12\end{array}\right|\left|\begin{array}{cc}0& 0\\ 0& 0\end{array}\right|=0$
Hence, $A\left(adjA\right)=\left(adjA\right)A=|A|I_2$.