Definition:Echelon Matrix

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

Row Echelon Form
The matrix $\mathbf A$ is in row echelon form if:
 * $(1): \quad$ The leading coefficient in each non-zero row is $1$
 * $(2): \quad$ The leading $1$ in any non-zero row occurs to the right of the leading $1$ in any previous row
 * $(3): \quad$ The non-zero rows appear before any zero rows.

Such a matrix is called an echelon matrix.

Reduced Row Echelon Form
An echelon matrix is in reduced row echelon form if, in addition, the leading $1$ in any non-zero row is the only non-zero element in the column in which that $1$ occurs.

This is also called row canonical form.

Such a matrix is called a reduced echelon 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
Some sources do not require that, for a matrix to be in row echelon form, the first non-zero element in each non-zero row must be $1$.

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