Answer
the inverse of the matrix $A$ is
$A^{-1}=\left[\begin{array}{ccccccc}
1 & 0 &1/4 \\
0 & 1/6& 1/24 \\
0 & 0 &1/4
\end{array}\right] \quad $
Work Step by Step
the augmented matrix
$\left[\begin{array}{ccccccc}
1 & 0 &-1 & | 1 & 0 &0 \\
0 & 6& -1& | 0 & 1& 0 \\
0 & 0 & 4 & | 0 & 0 &1
\end{array}\right] \quad $
by using $\quad R_{2} / 6 \rightarrow R_{2}$
$\left[\begin{array}{ccccccc}
1 & 0 &-1 & | 1 & 0 &0 \\
0 & 1& -1/6& | 0 & 1/6& 0 \\
0 & 0 & 4 & | 0 & 0 &1
\end{array}\right] \quad $
Using the relation $R_{3} / 4 \rightarrow R_{3}$
$\left[\begin{array}{ccccccc}
1 & 0 &-1 & | 1 & 0 &0 \\
0 & 1& -1/6& | 0 & 1/6& 0 \\
0 & 0 & 1 & | 0 & 0 &1/4
\end{array}\right] \quad $
Using the relation $R_{1} + R_{3} \rightarrow R_{1}$ , we have
$\left[\begin{array}{ccccccc}
1 & 0 &0 & | 1 & 0 &1/4 \\
0 & 1& -1/6& | 0 & 1/6& 0 \\
0 & 0 & 1 & | 0 & 0 &1/4
\end{array}\right] \quad $
using the relation
$R_{2} + R_{3}/6 \rightarrow R_{2}$
$\left[\begin{array}{ccccccc}
1 & 0 &0 & | 1 & 0 &1/4 \\
0 & 1& 0/6& | 0 & 1/6& 1/24 \\
0 & 0 & 1 & | 0 & 0 &1/4
\end{array}\right] \quad $
thus the inverse of the matrix $A$ is
$A^{-1}=\left[\begin{array}{ccccccc}
1 & 0 &1/4 \\
0 & 1/6& 1/24 \\
0 & 0 &1/4
\end{array}\right] \quad $
