REF of $$$\left[\begin{array}{ccccc}1 & 3 & -3 & -2 & -1\\2 & -2 & 2 & 4 & -2\\-1 & -2 & 2 & 1 & 1\end{array}\right]$$$

Find the row echelon form of $$$\left[\begin{array}{ccccc}1 & 3 & -3 & -2 & -1\\2 & -2 & 2 & 4 & -2\\-1 & -2 & 2 & 1 & 1\end{array}\right]$$$.

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

$$$\left[\begin{array}{ccccc}1 & 3 & -3 & -2 & -1\\0 & -8 & 8 & 8 & 0\\-1 & -2 & 2 & 1 & 1\end{array}\right]$$$

Add row $$$1$$$ to row $$$3$$$: $$$R_{3} = R_{3} + R_{1}$$$.

$$$\left[\begin{array}{ccccc}1 & 3 & -3 & -2 & -1\\0 & -8 & 8 & 8 & 0\\0 & 1 & -1 & -1 & 0\end{array}\right]$$$

Add row $$$2$$$ multiplied by $$$\frac{1}{8}$$$ to row $$$3$$$: $$$R_{3} = R_{3} + \frac{R_{2}}{8}$$$.

$$$\left[\begin{array}{ccccc}1 & 3 & -3 & -2 & -1\\0 & -8 & 8 & 8 & 0\\0 & 0 & 0 & 0 & 0\end{array}\right]$$$

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. Move to the next column.

Since the element at row $$$3$$$ 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 $$$3$$$ 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 row echelon form is $$$\left[\begin{array}{ccccc}1 & 3 & -3 & -2 & -1\\0 & -8 & 8 & 8 & 0\\0 & 0 & 0 & 0 & 0\end{array}\right]$$$A.