$$\left[\begin{matrix} 2 & -1 & 1 \\-1 & 2 & -1 \\1 & -1 & 2 \end{matrix}\right] { A }^{ 3 } -6 { A }^{ 2 } +9A-4I=0$$
$A^{3}-6A^{2}+9A-4I=0$
$\left[\begin{array}{rrr}3 & 1 & -1 \\ 1 & 3 & 1 \\ -1 & 1 & 3\end{array}\right]$
$A_{11}=\left|\begin{array}{rr}2 & -1 \\ -1 & 2\end{array}\right|=3$
$A_{12}=-\left|\begin{array}{rr}-1 & -1 \\ 1 & 2\end{array}\right|=1$
$A_{13}=\left|\begin{array}{rr}-1 & 2 \\ 1 & -1\end{array}\right|=-1$
$A_{21}=-\left|\begin{array}{rr}-1 & 1 \\ -1 & 2\end{array}\right|=1$
$A_{22}=\left|\begin{array}{rr}2 & 1 \\ 1 & 2\end{array}\right|=3$
$A_{23}=-\left|\begin{array}{rr}2 & -1 \\ 1 & -1\end{array}\right|=1$
$A_{31}=\left|\begin{array}{rr}-1 & 1 \\ 2 & -1\end{array}\right|=-1$
$A_{32}=-\left|\begin{array}{rr}2 & 1 \\ -1 & -1\end{array}\right|=1$
$A_{33}=\left|\begin{array}{rr}2 & -1 \\ -1 & 2\end{array}\right|=3$
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