Gauss-Jordan elimination on $$$\left[\begin{array}{ccc|c}3 & 3 & 2 & 3\\9 & 5 & 8 & 7\\3 & 1 & 3 & 2\end{array}\right]$$$

Perform the Gauss-Jordan elimination (reduce completely) on $$$\left[\begin{array}{cccc}3 & 3 & 2 & 3\\9 & 5 & 8 & 7\\3 & 1 & 3 & 2\end{array}\right]$$$.

Divide row $$$1$$$ by $$$3$$$: $$$R_{1} = \frac{R_{1}}{3}$$$.

$$$\left[\begin{array}{ccc|c}1 & 1 & \frac{2}{3} & 1\\9 & 5 & 8 & 7\\3 & 1 & 3 & 2\end{array}\right]$$$

Subtract row $$$1$$$ multiplied by $$$9$$$ from row $$$2$$$: $$$R_{2} = R_{2} - 9 R_{1}$$$.

$$$\left[\begin{array}{ccc|c}1 & 1 & \frac{2}{3} & 1\\0 & -4 & 2 & -2\\3 & 1 & 3 & 2\end{array}\right]$$$

Subtract row $$$1$$$ multiplied by $$$3$$$ from row $$$3$$$: $$$R_{3} = R_{3} - 3 R_{1}$$$.

$$$\left[\begin{array}{ccc|c}1 & 1 & \frac{2}{3} & 1\\0 & -4 & 2 & -2\\0 & -2 & 1 & -1\end{array}\right]$$$

Divide row $$$2$$$ by $$$-4$$$: $$$R_{2} = - \frac{R_{2}}{4}$$$.

$$$\left[\begin{array}{ccc|c}1 & 1 & \frac{2}{3} & 1\\0 & 1 & - \frac{1}{2} & \frac{1}{2}\\0 & -2 & 1 & -1\end{array}\right]$$$

Subtract row $$$2$$$ from row $$$1$$$: $$$R_{1} = R_{1} - R_{2}$$$.

$$$\left[\begin{array}{ccc|c}1 & 0 & \frac{7}{6} & \frac{1}{2}\\0 & 1 & - \frac{1}{2} & \frac{1}{2}\\0 & -2 & 1 & -1\end{array}\right]$$$

Add row $$$2$$$ multiplied by $$$2$$$ to row $$$3$$$: $$$R_{3} = R_{3} + 2 R_{2}$$$.

$$$\left[\begin{array}{ccc|c}1 & 0 & \frac{7}{6} & \frac{1}{2}\\0 & 1 & - \frac{1}{2} & \frac{1}{2}\\0 & 0 & 0 & 0\end{array}\right]$$$

Since the element at row $$$3$$$ and column $$$3$$$ (pivot element) equals $$$0$$$, we need to swap the rows.

Find the first nonzero element in column $$$3$$$ under the pivot entry.

As can be seen, there are no such entries.

The reduced matrix is $$$\left[\begin{array}{cccc}1 & 0 & \frac{7}{6} & \frac{1}{2}\\0 & 1 & - \frac{1}{2} & \frac{1}{2}\\0 & 0 & 0 & 0\end{array}\right]\approx \left[\begin{array}{cccc}1 & 0 & 1.166666666666667 & 0.5\\0 & 1 & -0.5 & 0.5\\0 & 0 & 0 & 0\end{array}\right].$$$A