$A=\left[\begin{array}{ccc}1& -1& 1\\ 2& -1& 0\\ 1& 0& 0\end{array}\right]$

$A^2=\left[\begin{array}{ccc}1& -1& 1\\ 2& -1& 0\\ 1& 0& 0\end{array}\right]\left[\begin{array}{ccc}1& -1& 1\\ 2& -1& 0\\ 1& 0& 0\end{array}\right]$

$\Rightarrow A^2=\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 2\\ 1& -1& 1\end{array}\right]$

Now, $A^3=\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 2\\ 1& -1& 1\end{array}\right]\left[\begin{array}{ccc}1& -1& 1\\ 2& -1& 0\\ 1& 0& 0\end{array}\right]$, $\left\{\because A^3=A^2.A\right\}$

$\Rightarrow A^3=\left[\begin{array}{ccc}1& 0& 0\\ 4& -1& 0\\ 1& 0& 1\end{array}\right]$

Now, $A^6=A^3.A^3$

$\Rightarrow A^6=\left[\begin{array}{ccc}1& 0& 0\\ 4& -1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 4& -1& 0\\ 0& 0& 1\end{array}\right]$

$\Rightarrow A^6=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Also, $\left|A^6\right|=1$

$\Rightarrow \left|A\right|\ne 0$

So, $A$ is non-singular.

$\Rightarrow A^6=I$

Also, $A{A}^{-1}=A^{-1}A=I$

$\Rightarrow A^{-1}{A}^6=A^{-1}I$

$\Rightarrow A^5=A^{-1}$