Inverse of $$$\left[\begin{array}{ccc}1 & 1 & 1\\2 & 3 & 4\\3 & 1 & 1\end{array}\right]$$$
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Calculate $$$\left[\begin{array}{ccc}1 & 1 & 1\\2 & 3 & 4\\3 & 1 & 1\end{array}\right]^{-1}$$$ using the Gauss-Jordan elimination.
Solution
To find the inverse matrix, augment it with the identity matrix and perform row operations trying to make the identity matrix to the left. Then to the right will be the inverse matrix.
So, augment the matrix with the identity matrix:
$$$\left[\begin{array}{ccc|ccc}1 & 1 & 1 & 1 & 0 & 0\\2 & 3 & 4 & 0 & 1 & 0\\3 & 1 & 1 & 0 & 0 & 1\end{array}\right]$$$
Subtract row $$$1$$$ multiplied by $$$2$$$ from row $$$2$$$: $$$R_{2} = R_{2} - 2 R_{1}$$$.
$$$\left[\begin{array}{ccc|ccc}1 & 1 & 1 & 1 & 0 & 0\\0 & 1 & 2 & -2 & 1 & 0\\3 & 1 & 1 & 0 & 0 & 1\end{array}\right]$$$
Subtract row $$$1$$$ multiplied by $$$3$$$ from row $$$3$$$: $$$R_{3} = R_{3} - 3 R_{1}$$$.
$$$\left[\begin{array}{ccc|ccc}1 & 1 & 1 & 1 & 0 & 0\\0 & 1 & 2 & -2 & 1 & 0\\0 & -2 & -2 & -3 & 0 & 1\end{array}\right]$$$
Subtract row $$$2$$$ from row $$$1$$$: $$$R_{1} = R_{1} - R_{2}$$$.
$$$\left[\begin{array}{ccc|ccc}1 & 0 & -1 & 3 & -1 & 0\\0 & 1 & 2 & -2 & 1 & 0\\0 & -2 & -2 & -3 & 0 & 1\end{array}\right]$$$
Add row $$$2$$$ multiplied by $$$2$$$ to row $$$3$$$: $$$R_{3} = R_{3} + 2 R_{2}$$$.
$$$\left[\begin{array}{ccc|ccc}1 & 0 & -1 & 3 & -1 & 0\\0 & 1 & 2 & -2 & 1 & 0\\0 & 0 & 2 & -7 & 2 & 1\end{array}\right]$$$
Divide row $$$3$$$ by $$$2$$$: $$$R_{3} = \frac{R_{3}}{2}$$$.
$$$\left[\begin{array}{ccc|ccc}1 & 0 & -1 & 3 & -1 & 0\\0 & 1 & 2 & -2 & 1 & 0\\0 & 0 & 1 & - \frac{7}{2} & 1 & \frac{1}{2}\end{array}\right]$$$
Add row $$$3$$$ to row $$$1$$$: $$$R_{1} = R_{1} + R_{3}$$$.
$$$\left[\begin{array}{ccc|ccc}1 & 0 & 0 & - \frac{1}{2} & 0 & \frac{1}{2}\\0 & 1 & 2 & -2 & 1 & 0\\0 & 0 & 1 & - \frac{7}{2} & 1 & \frac{1}{2}\end{array}\right]$$$
Subtract row $$$3$$$ multiplied by $$$2$$$ from row $$$2$$$: $$$R_{2} = R_{2} - 2 R_{3}$$$.
$$$\left[\begin{array}{ccc|ccc}1 & 0 & 0 & - \frac{1}{2} & 0 & \frac{1}{2}\\0 & 1 & 0 & 5 & -1 & -1\\0 & 0 & 1 & - \frac{7}{2} & 1 & \frac{1}{2}\end{array}\right]$$$
We are done. On the left is the identity matrix. On the right is the inverse matrix.
Answer
The inverse matrix is $$$\left[\begin{array}{ccc}- \frac{1}{2} & 0 & \frac{1}{2}\\5 & -1 & -1\\- \frac{7}{2} & 1 & \frac{1}{2}\end{array}\right] = \left[\begin{array}{ccc}-0.5 & 0 & 0.5\\5 & -1 & -1\\-3.5 & 1 & 0.5\end{array}\right].$$$A