Definition of reduced row echelon form of a matrix
Created almost 9 years ago, updated 5 months ago
ID23544
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.
**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.