<s>[INST] How to invert these matrix via Gauss method? Im trying to find the inverse matrix based on
$\left[\begin{array}{ccc}
2 & 2 & -1\\
0 & 4 & -1\\
-1 & -2 & 1
\end{array}\right]$
$R_{1}=$ First row
$R_{2}=$ Second row
$R_{3}=$Third row
but at a certain step I can't reduce it further. Which must be the next operation?
$\left[\begin{array}{ccc}
2 & 2 & -1\\
0 & 4 & -1\\
-1 & -2 & 1
\end{array}\mid\begin{array}{ccc}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{array}\right]$ $\rightarrow\frac{R_{1}}{2}$
$\left[\begin{array}{ccc}
1 & 1 & \frac{-1}{2}\\
0 & 4 & -1\\
-1 & -2 & 1
\end{array}\mid\begin{array}{ccc}
\frac{1}{2} & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{array}\right]$ $\rightarrow R_{3}+R_{1}$
$\left[\begin{array}{ccc}
1 & 1 & \frac{-1}{2}\\
0 & 4 & -1\\
0 & -1 & \frac{1}{2}
\end{array}\mid\begin{array}{ccc}
\frac{1}{2} & 0 & 0\\
0 & 1 & 0\\
\frac{1}{2} & 0 & 1
\end{array}\right]$$\rightarrow\frac{R_{2}}{4}$
$\left[\begin{array}{ccc}
1 & 1 & \frac{-1}{2}\\
0 & 1 & \frac{-1}{4}\\
0 & -1 & \frac{1}{2}
\end{array}\mid\begin{array}{ccc}
\frac{1}{2} & 0 & 0\\
0 & \frac{1}{4} & 0\\
\frac{1}{2} & 0 & 1
\end{array}\right]$$\rightarrow R_{1}-R_{2}$
$\left[\begin{array}{ccc}
1 & 0 & \frac{-1}{4}\\
0 & 1 & \frac{-1}{4}\\
0 & -1 & \frac{1}{2}
\end{array}\mid\begin{array}{ccc}
\frac{1}{2} & -\frac{1}{4} & 0\\
0 & \frac{1}{4} & 0\\
\frac{1}{2} & 0 & 1
\end{array}\right]$$\rightarrow R_{3}+R_{2}$
$\left[\begin{array}{ccc}
1 & 0 & \frac{-1}{4}\\
0 & 1 & \frac{-1}{4}\\
0 & 0 & \frac{1}{4}
\end{array}\mid\begin{array}{ccc}
\frac{1}{2} & -\frac{1}{4} & 0\\
0 & \frac{1}{4} & 0\\
\frac{1}{2} & \frac{1}{4} & 1
\end{array}\right]$$\rightarrow4R_{3}$
$\left[\begin{array}{ccc}
1 & 0 & \frac{-1}{4}\\
0 & 1 & \frac{-1}{4}\\
0 & 0 & 1
\end{array}\mid\begin{array}{ccc}
\frac{1}{2} & -\frac{1}{4} & 0\\
0 & \frac{1}{4} & 0\\
2 & 1 & 4
\end{array}\right]$$\rightarrow?$
 [/INST] Continuing with $R_{1}\rightarrow R_{1}+\frac{1}{4}R_3$ and $R_{2}\rightarrow R_{2}+\frac{1}{4}R_{3}$ we have
$$\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{array}\mid\begin{array}{ccc}
1 & 0 & 1\\
\frac{1}{2} & \frac{1}{2} & 1\\
2 & 1 & 4
\end{array}\right]$$
</s>