\begin{definition}[Definition:Echelon Matrix/Reduced Echelon Form]
Let $\mathbf A = \sqbrk a_{m n}$ be a matrix in echelon form whose order is $m \times n$.
The matrix $\mathbf A$ is in '''reduced echelon form''' {{iff}}, in addition to being in echelon form, the leading $1$ in any non-zero row is the only non-zero element in the column in which that $1$ occurs.
Such a matrix is called a '''reduced echelon matrix'''.
\end{definition}