[Math] Finding the inverse of a matrix by Gaussian elimination

Question

I spent last hours trying to figure out how to solve the inverse matrix to this matrix:
$$\begin{pmatrix}
2 &-3 & 1 \\
1 & 2 &-1 \\
2 & 1 & 1
\end{pmatrix}$$

The correct result should be
$$\begin{pmatrix}
0.250 & 0.333 & 0.083 \\
-0.250 & 0.000 & 0.250 \\
-0.250 & -0.667 & 0.583
\end{pmatrix}$$

However, I am still unable to get there. Here how I tried it (with using the Gaussian Elimination Rule):

$$\begin{multline}
\left(
\begin{array}{ccc|ccc}
2 & -3 & 1 & 1 & 0 & 0 \\
1 & 2 & -1 & 0 & 1 & 0 \\
2 & 1 & 1 & 0 & 0 & 1
\end{array}
\right)
\overset{[1] – 2[2] \rightarrow [2]}{\Longrightarrow}
\left(
\begin{array}{ccc|ccc}
2 & -3 & 1 & 1 & 0 & 0 \\
0 & -7 & 2 & 1 & -2 & 0 \\
2 & 1 & 1 & 0 & 0 & 1
\end{array}
\right)
\overset{[1] – [3] \rightarrow [3]}{\Longrightarrow} \\
\left(
\begin{array}{ccc|ccc}
2 & -3 & 1 & 1 & 0 & 0 \\
0 & -7 & 2 & 1 & -2 & 0 \\
0 & -4 & 0 & 1 & 0 & -1
\end{array}
\right)
\overset{4[2] – 7[3] \rightarrow [3]}{\Longrightarrow}
\left(
\begin{array}{ccc|ccc}
2 & -3 & 1 & 1 & 0 & 0 \\
0 & -7 & 2 & 1 & -2 & 0 \\
0 & 0 & 8 & -3 & -8 & 7
\end{array}
\right)
\overset{4[2] – [3] \rightarrow [2]}{\Longrightarrow} \\
\left(
\begin{array}{ccc|ccc}
2 & -3 & 1 & 1 & 0 & 0 \\
0 & -28 & 0 & 7 & 0 & -7 \\
0 & 0 & 8 & -3 & -8 & 7
\end{array}
\right)
\overset{8[1] – 3[2] \rightarrow [1]}{\Longrightarrow}
\left(
\begin{array}{ccc|ccc}
16 & -24 & 0 & 11 & 8 & -7 \\
0 & -28 & 0 & 7 & 0 & -7 \\
0 & 0 & 8 & -3 & -8 & 7
\end{array}
\right)
\Longrightarrow \\
\left(
\begin{array}{ccc|ccc}
2 & -3 & 0 & \tfrac{11}{8} & 1 & \tfrac{-7}{8} \\
0 & -7 & 0 & \tfrac{7}{4} & 0 & \tfrac{-7}{4} \\
0 & 0 & 8 & -3 & -8 & 7
\end{array}
\right)
\overset{7[1] – 3[2] \rightarrow [1]}{\Longrightarrow}
\left(
\begin{array}{ccc|ccc}
14 & 0 & 0 & \tfrac{35}{8} & 7 & \tfrac{-7}{8} \\
0 & -7 & 0 & \tfrac{7}{4} & 0 & \tfrac{-7}{4} \\
0 & 0 & 8 & -3 & -8 & 7
\end{array}
\right)
\Longrightarrow \\
\left(
\begin{array}{ccc|ccc}
1 & 0 & 0 & 0.3125 & 0.5 & -0.0625 \\
0 & 1 & 0 & -0.25 & 0 & 0.25 \\
0 & 0 & 1 & -0.375 & -1 & 0.875
\end{array}
\right)
\end{multline}$$

(Original images: one, two)

I would be very grateful guys for helping me to figure out what I am doing wrong, there's always something why the whole inverse matrix is not correct.

Thank you very much

Answer 1

At first step [1]-2[2], i think the entry (2,3) is 3, while you wrote 2.