Question

If $A=\left[\begin{array}{ccc}1& -1& 2\\ 3& 0& -2\\ 1& 0& 3\end{array}\right]$ verify that $A\left(adjA\right)=\left|A\right|I$.

Answer 1

$\left|A\right|=+1(9+2)=11$

$adj\left(A\right)$=Transfer co factor matrix

$c_{11}=\left|\begin{array}{cc}0& -2\\ 0& 3\end{array}\right|=0,c_{23}=\left|\begin{array}{cc}1& -1\\ 1& 0\end{array}\right|=-1c_{12}=\left|\begin{array}{cc}3& -2\\ 1& 3\end{array}\right|=11,c_{31}=\left|\begin{array}{cc}-1& 2\\ 0& -2\end{array}\right|=2c_{13}=\left|\begin{array}{cc}3& 0\\ 1& 0\end{array}\right|=0,c_{32}=-\left|\begin{array}{cc}1& 2\\ 3& -2\end{array}\right|=8c_{21}=\left|\begin{array}{cc}-1& 2\\ 0& 3\end{array}\right|=3,c_{33}=\left|\begin{array}{cc}1& -1\\ 3& 0\end{array}\right|=3c_{22}=\left|\begin{array}{cc}1& 2\\ 1& 2\end{array}\right|=1$

Co factor matrix $\left|\begin{array}{ccc}0& -11& 0\\ 3& 1& -1\\ 2& 8& 3\end{array}\right|$

$adj\left(A\right)$=$\left|\begin{array}{ccc}0& 3& 2\\ -11& 1& 8\\ 0& -1& 3\end{array}\right|$

$A\left(adj\right(A\left)\right)$=$\left|\begin{array}{ccc}1& -1& 2\\ 3& 0& -2\\ 1& 0& 3\end{array}\right|\left|\begin{array}{ccc}0& 3& 2\\ -11& 1& 8\\ 0& -1& 3\end{array}\right|$

$\left|\begin{array}{ccc}11& 0& 0\\ 0& 11& 0\\ 0& 0& 11\end{array}\right|=\left|11\right|I=AI$