Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 3

Examples of Use of Matrix is Row Equivalent to Echelon Matrix
Let $\mathbf A = \begin {bmatrix} 1 & 2 & 3 & 5 \\ 1 & 2 & 3 & 4 \\ 0 & 0 & 1 & 1 \\ \end {bmatrix}$

This can be converted into the echelon form:
 * $\mathbf E = \begin {bmatrix} 1 & 2 & 3 & 5 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ \end {bmatrix}$

Proof
Using Row Operation to Clear First Column of Matrix we obtain:
 * $\mathbf B = \begin {bmatrix} 1 & 2 & 3 & 5 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 1 \\ \end {bmatrix}$

which is obtained by adding $-2$ of row $1$ to row $2$.

Then we investigate the submatrix:


 * $\mathbf B' = \begin {bmatrix} 0 & 0 & -1 \\ 0 & 1 & 1 \\ \end {bmatrix}$

Using Row Operation to Clear First Column of Matrix we obtain:
 * $\mathbf C' = \begin {bmatrix} 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end {bmatrix}$

which is obtained by:
 * $(1): \quad$ exchanging row $1$ of $\mathbf B'$ with row $2$ of $\mathbf B'$.
 * $(2): \quad$ multiplying row $2$ of $\mathbf B'$ by $-1$.

Thus we are left with:
 * $\mathbf E = \begin {bmatrix} 1 & 2 & 3 & 5 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ \end {bmatrix}$

Also presented as
Some sources use the non-unity variant of the echelon matrix.

Such sources do not require that the leading coefficients necessarily have to equal to $1$.

Hence they consider the final step to convert row $3$ of $\mathbf E$ from $\begin {bmatrix} 0 & 0 & 0 & -1 \end {bmatrix}$ to $\begin {bmatrix} 0 & 0 & 0 & 1 \end {bmatrix}$ as optional.