Transform any matrix to Row Echelon Form or Reduced Row Echelon Form and find its Rank — with every row operation explained step by step.
Matrix Setup
Results Summary
Row Echelon Form (REF)
Final REF Matrix
Step-by-Step Row Operations
Reduced Row Echelon Form (RREF)
Final RREF Matrix
Back-Substitution Steps (REF → RREF)
Rank Analysis
Theory & Definitions
Row Echelon Form (REF) — A matrix is in REF if:
All zero rows are at the bottom.
The leading entry (pivot) in each non-zero row is strictly to the right of the pivot in the row above.
All entries below a pivot are zero.
Reduced Row Echelon Form (RREF) — REF with two extra conditions:
Each pivot is 1 (scaled).
All entries above and below each pivot are 0.
RREF is unique — every matrix has exactly one RREF.
Rank — The rank of a matrix is the number of non-zero rows in its REF (= number of pivot positions).
$$\text{rank}(A) = \text{number of pivots in REF/RREF}$$
Also: $\text{rank}(A) + \text{nullity}(A) = n$ (number of columns).