Definition:Echelon Matrix/Echelon Form/Non-Unity Variant

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

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.

Also defined as
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 $1$.