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

Theorem: Every matrix is row-equivalent to a matrix in reduced row echelon form.

Proof: The Gauss-Jordan elimination algorithm always terminates and produces a matrix in RREF. The algorithm works because:

1. At each step, if there is a nonzero entry in the current column below the current row, we can always find a pivot (by row swapping if needed).
2. Row replacement operations eliminate all entries below the pivot.
3. After forward elimination, we scale each pivot to 1.
4. Backward elimination clears all entries above each pivot.

Since each step involves a finite number of operations and the number of rows and columns is finite, the process terminates. The result satisfies all four RREF conditions by construction.