$\begin{aligned} \text { We have } A &=\left[\begin{array}{ccc}2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2\end{array}\right] \\ \mathrm{A}^2=\mathrm{AA}=& {\left[\begin{array}{ccc}2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2\end{array}\right]\left[\begin{array}{ccc}2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2\end{array}\right] } \\ &=\left[\begin{array}{ccc}6 & -5 & 5 \\ -5 & 6 & -5 \\ 5 & -5 & 6\end{array}\right] \\ \mathrm{A}^3=\mathrm{A}^2 \mathrm{~A} &=\left[\begin{array}{ccc}6 & -5 & 5 \\ -5 & 6 & -5 \\ 5 & -5 & 6\end{array}\right]\left[\begin{array}{ccc}2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2\end{array}\right] \end{aligned}$
$=\left[\begin{array}{ccc}
22 & -21 & 21 \\
-21 & 22 & -21 \\
21 & -21 & 22
\end{array}\right]$
Now, $\mathrm{A}^3-6 \mathrm{~A}^2+9 \mathrm{~A}-4 \mathrm{I}$
$=\left[\begin{array}{ccc}22 & -21 & 21 \\ -21 & 22 & -21 \\ 21 & -21 & 22\end{array}\right]-6\left[\begin{array}{ccc}6 & -5 & 5 \\ -5 & 6 & -5 \\ 5 & -5 & 6\end{array}\right]$
$+9\left[\begin{array}{ccc}2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2\end{array}\right]-4\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]=0$
Hence, $A^3-6 A^2+9 A-4 I=0$
$\Rightarrow \quad 4 \mathrm{I}=\mathrm{A}^3-6 \mathrm{~A}^2+9 \mathrm{~A}$
$\Rightarrow \quad \mathrm{A}^{-1}=\frac{1}{4} \mathrm{~A}^2-\frac{6}{4} \mathrm{~A}+\frac{9}{4} \mathrm{I}$
$=\frac{1}{4}\left[\begin{array}{ccc}6 & -5 & 5 \\ -5 & 6 & -5 \\ 5 & -5 & 6\end{array}\right]-\frac{6}{4}\left[\begin{array}{ccc}2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2\end{array}\right]$
$+\frac{9}{4}\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]=\frac{1}{4}\left[\begin{array}{ccc}3 & 1 & -1 \\ 1 & 3 & 1 \\ -1 & 1 & 3\end{array}\right]$