$\begin{aligned} & {\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{array}\right] \mathrm{A}^{-1}=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]} \\ & \mathrm{R}_1 \rightarrow \mathrm{R}_1+\mathrm{R}_2 \text { and } \mathrm{R}_3 \rightarrow \mathrm{R}_3-3 \mathrm{R}_2 \\ & {\left[\begin{array}{ccc}1 & 3 & 5 \\ 1 & 2 & 3 \\ 0 & -5 & -8\end{array}\right] \mathrm{A}^{-1}=\left[\begin{array}{ccc}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & -3 & 1\end{array}\right]} \\ & \mathrm{R}_2 \rightarrow \mathrm{R}_2-\mathrm{R}_1 \\ & {\left[\begin{array}{ccc}1 & 3 & 5 \\ 0 & -1 & -2 \\ 0 & -5 & -8\end{array}\right] \mathrm{A}^{-1}=\left[\begin{array}{ccc}1 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & -3 & 1\end{array}\right]} \\ & \mathrm{R}_1 \rightarrow \mathrm{R}_1+3 \mathrm{R}_2 \text { and } \mathrm{R}_3 \rightarrow \mathrm{R}_3-5 \mathrm{R}_2 \\ & {\left[\begin{array}{ccc}1 & 0 & -1 \\ 0 & -1 & -2 \\ 0 & 0 & 2\end{array}\right] \mathrm{A}^{-1}=\left[\begin{array}{ccc}-2 & 1 & 0 \\ -1 & 0 & 0 \\ 5 & -3 & 1\end{array}\right]}\end{aligned}$
$\mathrm{R}_2 \rightarrow \mathrm{R}_2+\mathrm{R}_3$ and $\mathrm{R}_1 \rightarrow 2 \mathrm{R}_1+\mathrm{R}_3$
$\begin{aligned} & {\left[\begin{array}{ccc}2 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 2\end{array}\right] \mathrm{A}^{-1}=\left[\begin{array}{ccc}1 & -1 & 1 \\ 4 & -3 & 1 \\ 5 & -3 & 1\end{array}\right]} \\ & \mathrm{R}_1=\frac{1}{2} \mathrm{R}_1, \mathrm{R}_2 \rightarrow-\mathrm{R}_2, \mathrm{R}_3 \rightarrow \frac{1}{2} \mathrm{R}_3 \\ & {\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right] \mathrm{A}^{-1}=\left[\begin{array}{ccc}\frac{1}{2} & \frac{-1}{2} & \frac{1}{2} \\ -4 & 3 & -1 \\ \frac{5}{2} & \frac{-3}{2} & \frac{1}{2}\end{array}\right]}\end{aligned}$