Reduced row echelon form

This article defines a property that can be evaluated for a matrix. In other words, given a matrix, the matrix either satisfies or does not satisfy the property.
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Definition

A matrix being in reduced row echelon form

A matrix is said to be a reduced row echelon matrix, or said to be in reduced row echelon form (rref), if it satisfies the following conditions:

• All nonzero rows are above all zero rows. Here, a nonzero row is a row that has at least one nonzero entry, and a zero row is a row where all entries are zero.
• The first nonzero entry in any nonzero row occurs in a strictly later column than the first nonzero entry in the row immediately above it (and hence also, in all the rows above it).
• The first nonzero entry in any nonzero row is 1 (this condition is omitted in some definitions) and all other entries in the column of that entry are zero.

With the exception of the (emphasized) second half of the last condition, the conditions above define row echelon form.

The reduced row echelon form of a matrix

For a $m\times n$ matrix </math>A</math>, the reduced row echelon form of $A$ is the unique matrix $B$ in reduced row echelon form such that we can write $B=SA$ where $S$ is a $m\times m$ invertible matrix.