# Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 4

## Examples of Use of Matrix is Row Equivalent to Echelon Matrix

Let $\mathbf A = \begin {bmatrix} 1 & 1 & 2 & 3 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 3 & 3 \\ \end {bmatrix}$

This can be converted into the echelon form:

$\mathbf E = \begin {bmatrix} 1 & 1 & 2 & 3 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ \end {bmatrix}$

## Proof

It is noted that $\mathbf A$ is already most of the way there.

It remains to use the elementary row operation:

$e := r_3 \to r_3 - 3 r_2$

to convert $\mathbf A$ to the form:

$\mathbf E = \begin {bmatrix} 1 & 1 & 2 & 3 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ \end {bmatrix}$

$\blacksquare$