Question

$\left|\begin{array}{ccc}1& -1& 2\\ 3& 0& 2\\ 1& 0& 2\end{array}\right|$

Answer 1

$M_{11}=0$ $M_{12}=11$ $M_{13}=0$ $M_{21}=0$ $M_{22}=1$ $M_{23}=1$ $M_{31}=2$ $M_{32}=-8$ $M_{33}=-3$ $C_{11}=0$ $C_{12}=11$ $C_{13}=0$ $C_{21}=3$ $C_{22}=1$ $C_{23}=-1$ $C_{31}=2$ $C_{32}=8$ $C_{33}=3$

co factor matrix $\left(\begin{array}{ccc}0& -11& 0\\ 3& 1& -1\\ +2& 8& 3\end{array}\right)$

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

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

$=\left(\begin{array}{ccc}11& 0& 0\\ 0& 11& 0\\ 0& 0& 11\end{array}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)=11I=\left|A\right|I$

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

$=11\left(\begin{array}{ccc}11& 0& 0\\ 0& 11& 0\\ 0& 0& 11\end{array}\right)=11\times I=\left|A\right|I$

$A.AdjA=Adj.A.A=\left|A\right|I$