Definition:Echelon Matrix

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) The leading coefficient in each non-zero row is $$1$$;
 * 2) The leading $$1$$ in any non-zero row occurs to the right of the leading 1 in any previous row;
 * 3) 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.