Definition:Echelon Matrix

Definition
Let $\mathbf A = \left[{a}\right]_{m n}$ be an $m \times n$ matrix.

Examples

 * $\begin{bmatrix}

1 & 5 & 4 & 2 \\ 0 & 0 & 1 & 7 \\ 0 & 1 & 0 & 9 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} $ is not an echelon matrix, because the leading $1$ in row $3$ occurs to the left of the leading $1$ in row $2$.


 * $\begin{bmatrix}

1 & 5 & 4 & 2 \\ 0 & 6 & 0 & 9 \\ 0 & 0 & 1 & 7 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} $ is not an echelon matrix, because the leading coefficient of row $2$ is not $1$.


 * $\begin{bmatrix}

1 & 5 & 4 & 2 \\ 0 & 1 & 0 & 9 \\ 0 & 0 & 1 & 7 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} $ is an echelon matrix, but not a reduced echelon matrix.


 * $\begin{bmatrix}

1 & 5 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} $ is a reduced echelon matrix.

Note
Although this definition refers to row echelon form, an equivalent definition for columns does not seem to be used.