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The inverse matrix of $\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{array}\right]$ is
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$\left[\begin{array}{ccc}\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \\ -4 & 3 & -1 \\ \frac{5}{2} & \frac{-3}{2} & \frac{1}{2}\end{array}\right]$
$A^{-1}=\frac{\operatorname{adj}(A)}{|A|}=\frac{1}{|A|} \cdot \operatorname{adj}(A)$
$A=\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{array}\right] ;|A|=0-1(1-9)+2(1-6)=8-10$
$|A|=-2 \neq 0$
$\operatorname{Adj} A=\left[\begin{array}{lll}A_{11} & A_{21} & A_{31} \\ A_{12} & A_{22} & A_{32} \\ A_{13} & A_{23} & A_{33}\end{array}\right]$
$A_{11}=(-1)^{1+1}[(2)(1)-(3)(1)]=-1$
$A_{12}=8, \quad A_{13}=-5, A_{21}=1, A_{22}=-6$
$A_{23}=3, A_{31}=-1, A_{32}=2, A_{33}=-1$
$\therefore A^{-1}=\frac{1}{-2}\left[\begin{array}{ccc}-1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1\end{array}\right]=\left[\begin{array}{ccc}1 / 2 & -1 / 2 & 1 / 2 \\ -4 & 3 & -1 \\ 5 / 2 & -3 / 2 & 1 / 2\end{array}\right]$.
$A=\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{array}\right] ;|A|=0-1(1-9)+2(1-6)=8-10$
$|A|=-2 \neq 0$
$\operatorname{Adj} A=\left[\begin{array}{lll}A_{11} & A_{21} & A_{31} \\ A_{12} & A_{22} & A_{32} \\ A_{13} & A_{23} & A_{33}\end{array}\right]$
$A_{11}=(-1)^{1+1}[(2)(1)-(3)(1)]=-1$
$A_{12}=8, \quad A_{13}=-5, A_{21}=1, A_{22}=-6$
$A_{23}=3, A_{31}=-1, A_{32}=2, A_{33}=-1$
$\therefore A^{-1}=\frac{1}{-2}\left[\begin{array}{ccc}-1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1\end{array}\right]=\left[\begin{array}{ccc}1 / 2 & -1 / 2 & 1 / 2 \\ -4 & 3 & -1 \\ 5 / 2 & -3 / 2 & 1 / 2\end{array}\right]$.
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