REF of $$$\left[\begin{array}{ccccc}1 & 3 & -3 & 5 & -1\\2 & 5 & 2 & 4 & 5\\-1 & -2 & -5 & 1 & 1\end{array}\right]$$$

Find the row echelon form of $$$\left[\begin{array}{ccccc}1 & 3 & -3 & 5 & -1\\2 & 5 & 2 & 4 & 5\\-1 & -2 & -5 & 1 & 1\end{array}\right]$$$.

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

$$$\left[\begin{array}{ccccc}1 & 3 & -3 & 5 & -1\\0 & -1 & 8 & -6 & 7\\-1 & -2 & -5 & 1 & 1\end{array}\right]$$$

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

$$$\left[\begin{array}{ccccc}1 & 3 & -3 & 5 & -1\\0 & -1 & 8 & -6 & 7\\0 & 1 & -8 & 6 & 0\end{array}\right]$$$

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

$$$\left[\begin{array}{ccccc}1 & 3 & -3 & 5 & -1\\0 & -1 & 8 & -6 & 7\\0 & 0 & 0 & 0 & 7\end{array}\right]$$$

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.

The row echelon form is $$$\left[\begin{array}{ccccc}1 & 3 & -3 & 5 & -1\\0 & -1 & 8 & -6 & 7\\0 & 0 & 0 & 0 & 7\end{array}\right]$$$A.