Every matrix is row-equivalent to only one matrix in reduced row echelon form.

Theorem (Uniqueness of RREF): Every matrix is row-equivalent to exactly one matrix in reduced row echelon form.

Proof sketch: Suppose a matrix $A$ is row-equivalent to two different RREF matrices $R_1$ and $R_2$. Then $R_1$ and $R_2$ are row-equivalent to each other, meaning they have the same row space. But the RREF of a matrix is uniquely determined by its row space: the pivot positions are determined by which columns are linearly independent, and the non-pivot columns are uniquely determined as linear combinations of pivot columns. Therefore $R_1 = R_2$.

Corollary: Two matrices are row-equivalent if and only if they have the same RREF. This provides an algorithm to test row equivalence.