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:


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