49. RREF
A matrix is in Reduced Row Echelon Form (RREF) if it satisfies all of the following:
-
Echelon form — see REF:
- Any all-zero rows are at the bottom
- Leading entries (pivots) move strictly right as you go down
- Each pivot is (the leading entry of every non-zero row)
- Each pivot is the only non-zero entry in its column (zeros above and below)
Example
RREF:
Not RREF (the pivot in column 2 is , not , and column 1 has a non-zero above the pivot):
49.1. RREF vs REF
| Form | Pivots | Above pivots | |—|—|—| | REF | non-zero | unrestricted | | RREF | exactly | all |
REF comes from Gaussian elimination RREF is what Gauss–Jordan elimination produces.
49.2. Uniqueness
Given a matrix , its REF is not unique (depends on row operations chosen), but its RREF is unique. So RREF is a canonical form — useful for:
- Reading off the rank: number of non-zero rows
- Reading off the solution to directly from in RREF
- Identifying linearly independent columns: the pivot columns
- Computing the null space basis from free variables
49.3. Reading the solution
After reducing augmented matrix to RREF:
Example
Pivot columns: 1, 2, 4 → pivot variables . Free columns: 3, 5 → free variables .
Solution (parametric):