48. Row Echelon Form

A matrix is in Row Echelon Form (REF) if it satisfies all of:

  1. Pivot — each non-zero row has a leading non-zero entry called a pivot
  2. Zeros below pivots — each pivot has zeros below it in its column
  3. Rightward staircase — each pivot is strictly to the right of the pivot in the row above
  4. Zero rows at bottom — any all-zero rows appear at the bottom

48.1. Elementary row operations

The three operations that don’t change the solution set of a system, used to drive a matrix toward REF:

  1. Row swap: exchange two rows
  2. Row scale: multiply a row by a non-zero scalar
  3. Row replacement: add a multiple of one row to another

These are applied via Gaussian elimination.

Example

System:

Augmented matrix:

Step 1: :

Step 2: , :

Step 3: :

Step 4: :

Step 5: (now in REF):

Step 6: back-substitute.

48.2. REF vs RREF

REF is not unique. The stronger RREF (Reduced Row Echelon Form) — pivots equal and zeros above as well as below — is unique.

48.3. See also