Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 5

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

This can be converted into the echelon form:
 * $\mathbf E = \begin {bmatrix} 1 & 0 & -1 & -2 & -3 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 0 & 0 \\ \end {bmatrix}$

Proof
This matrix can more easily be handled by direct application of elementary row operations, as follows.

Let $e_1$ be the elementary row operation:
 * $e_1 := r_3 \to r_3 - r_1$

which leaves: $\mathbf A_1 = \begin {bmatrix} -1 & 0 & 1 & 2 & 3 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & -2 & -4 & -6 & -8 \\ \end {bmatrix}$

Let $e_2$ be the elementary row operation:
 * $e_2 := r_3 \to r_3 + 2 r_2$

which leaves:
 * $\mathbf A_2 = \begin {bmatrix} -1 & 0 & 1 & 2 & 3 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 0 & 0 \\ \end {bmatrix}$

$\mathbf A_2$ is in non-unity echelon form.

It remains to perform the elementary row operation $e_3$ to convert it into echelon form:
 * $e_3 := r_1 \to -r_1$

which leaves:
 * $\mathbf E = \begin {bmatrix} 1 & 0 & -1 & -2 & -3 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 0 & 0 \\ \end {bmatrix}$

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 $1$ of $\mathbf E$ from $\begin {bmatrix} -1 & 0 & 1 & 2 & 3 \end {bmatrix}$ to $\begin {bmatrix} 1 & 0 & -1 & -2 & -3 \end {bmatrix}$ as optional.