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).
REF, RREF and rank: key ideas
In row echelon form (REF) every row of zeros is at the bottom and each leading entry lies to the right of the one above it. In reduced row echelon form (RREF) each leading entry is 1 and is the only non-zero entry in its column. The RREF of a matrix is unique, while a REF is not. The rank is the number of non-zero rows in REF, which equals the dimension of the row space and the column space. By the rank-nullity theorem, rank + nullity = number of columns.
Common questions
Is the rank the same from REF and RREF?
Yes. Both give the same number of pivots, so the same rank.
What does full rank mean?
The rank equals the smaller of the number of rows and columns.