Definition of reduced row echelon form of a matrix

Definition: The reduced row echelon form of a matrix $A$ is the unique matrix $R$ in RREF that is row-equivalent to $A$.

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

Why this matters: The RREF is a canonical form — it provides a unique "fingerprint" for the row space of a matrix. Two matrices have the same row space if and only if they have the same RREF.

Computation: The RREF is found by applying Gauss-Jordan elimination: forward elimination to reach REF, then scaling pivots to 1, then backward elimination to clear entries above pivots.