RREF of $$$\left[\begin{array}{ccccc}1 & 2 & 0 & -1 & 5\\2 & 0 & 2 & 0 & 1\\1 & 1 & -1 & 3 & 2\\0 & 3 & -3 & 2 & 6\end{array}\right]$$$

Find the reduced row echelon form of $$$\left[\begin{array}{ccccc}1 & 2 & 0 & -1 & 5\\2 & 0 & 2 & 0 & 1\\1 & 1 & -1 & 3 & 2\\0 & 3 & -3 & 2 & 6\end{array}\right]$$$.

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

$$$\left[\begin{array}{ccccc}1 & 2 & 0 & -1 & 5\\0 & -4 & 2 & 2 & -9\\1 & 1 & -1 & 3 & 2\\0 & 3 & -3 & 2 & 6\end{array}\right]$$$

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

$$$\left[\begin{array}{ccccc}1 & 2 & 0 & -1 & 5\\0 & -4 & 2 & 2 & -9\\0 & -1 & -1 & 4 & -3\\0 & 3 & -3 & 2 & 6\end{array}\right]$$$

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

$$$\left[\begin{array}{ccccc}1 & 2 & 0 & -1 & 5\\0 & 1 & - \frac{1}{2} & - \frac{1}{2} & \frac{9}{4}\\0 & -1 & -1 & 4 & -3\\0 & 3 & -3 & 2 & 6\end{array}\right]$$$

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

$$$\left[\begin{array}{ccccc}1 & 0 & 1 & 0 & \frac{1}{2}\\0 & 1 & - \frac{1}{2} & - \frac{1}{2} & \frac{9}{4}\\0 & -1 & -1 & 4 & -3\\0 & 3 & -3 & 2 & 6\end{array}\right]$$$

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

$$$\left[\begin{array}{ccccc}1 & 0 & 1 & 0 & \frac{1}{2}\\0 & 1 & - \frac{1}{2} & - \frac{1}{2} & \frac{9}{4}\\0 & 0 & - \frac{3}{2} & \frac{7}{2} & - \frac{3}{4}\\0 & 3 & -3 & 2 & 6\end{array}\right]$$$

Subtract row $$$2$$$ multiplied by $$$3$$$ from row $$$4$$$: $$$R_{4} = R_{4} - 3 R_{2}$$$.

$$$\left[\begin{array}{ccccc}1 & 0 & 1 & 0 & \frac{1}{2}\\0 & 1 & - \frac{1}{2} & - \frac{1}{2} & \frac{9}{4}\\0 & 0 & - \frac{3}{2} & \frac{7}{2} & - \frac{3}{4}\\0 & 0 & - \frac{3}{2} & \frac{7}{2} & - \frac{3}{4}\end{array}\right]$$$

Multiply row $$$3$$$ by $$$- \frac{2}{3}$$$: $$$R_{3} = - \frac{2 R_{3}}{3}$$$.

$$$\left[\begin{array}{ccccc}1 & 0 & 1 & 0 & \frac{1}{2}\\0 & 1 & - \frac{1}{2} & - \frac{1}{2} & \frac{9}{4}\\0 & 0 & 1 & - \frac{7}{3} & \frac{1}{2}\\0 & 0 & - \frac{3}{2} & \frac{7}{2} & - \frac{3}{4}\end{array}\right]$$$

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

$$$\left[\begin{array}{ccccc}1 & 0 & 0 & \frac{7}{3} & 0\\0 & 1 & - \frac{1}{2} & - \frac{1}{2} & \frac{9}{4}\\0 & 0 & 1 & - \frac{7}{3} & \frac{1}{2}\\0 & 0 & - \frac{3}{2} & \frac{7}{2} & - \frac{3}{4}\end{array}\right]$$$

Add row $$$3$$$ multiplied by $$$\frac{1}{2}$$$ to row $$$2$$$: $$$R_{2} = R_{2} + \frac{R_{3}}{2}$$$.

$$$\left[\begin{array}{ccccc}1 & 0 & 0 & \frac{7}{3} & 0\\0 & 1 & 0 & - \frac{5}{3} & \frac{5}{2}\\0 & 0 & 1 & - \frac{7}{3} & \frac{1}{2}\\0 & 0 & - \frac{3}{2} & \frac{7}{2} & - \frac{3}{4}\end{array}\right]$$$

Add row $$$3$$$ multiplied by $$$\frac{3}{2}$$$ to row $$$4$$$: $$$R_{4} = R_{4} + \frac{3 R_{3}}{2}$$$.

$$$\left[\begin{array}{ccccc}1 & 0 & 0 & \frac{7}{3} & 0\\0 & 1 & 0 & - \frac{5}{3} & \frac{5}{2}\\0 & 0 & 1 & - \frac{7}{3} & \frac{1}{2}\\0 & 0 & 0 & 0 & 0\end{array}\right]$$$

Since the element at row $$$4$$$ and column $$$4$$$ (pivot element) equals $$$0$$$, we need to swap the rows.

Find the first nonzero element in column $$$4$$$ under the pivot entry.

As can be seen, there are no such entries. Move to the next column.

Since the element at row $$$4$$$ and column $$$5$$$ (pivot element) equals $$$0$$$, we need to swap the rows.

Find the first nonzero element in column $$$5$$$ under the pivot entry.

As can be seen, there are no such entries.

The reduced row echelon form is $$$\left[\begin{array}{ccccc}1 & 0 & 0 & \frac{7}{3} & 0\\0 & 1 & 0 & - \frac{5}{3} & \frac{5}{2}\\0 & 0 & 1 & - \frac{7}{3} & \frac{1}{2}\\0 & 0 & 0 & 0 & 0\end{array}\right]\approx \left[\begin{array}{ccccc}1 & 0 & 0 & 2.333333333333333 & 0\\0 & 1 & 0 & -1.666666666666667 & 2.5\\0 & 0 & 1 & -2.333333333333333 & 0.5\\0 & 0 & 0 & 0 & 0\end{array}\right].$$$A