Matrix Rank Calculator
The rank is the number of independent rows: [[1, 2, 3], [2, 4, 6], [1, 0, 1]] has rank 2 because the second row is twice the first.
One row per line; separate entries with spaces or commas.
2
- Size
- 3 × 3
- Nullity (columns − rank)
- 1
- Full rank
- no
| col 1 | col 2 | col 3 | |
|---|---|---|---|
| row 1 | 1 | 0 | 1 |
| row 2 | 0 | 1 | 1 |
| row 3 | 0 | 0 | 0 |
Step-by-step solution
- Row-reduce the matrix (Gauss–Jordan elimination with partial pivoting)
- Count the non-zero rows: rank = 2
Matrix Rank Formula
rank = number of non-zero rows in echelon form
unique if rank A = rank [A|b] = n; none if rank A < rank [A|b]
Worked Example
The matrix [[1, 2, 3], [2, 4, 6], [1, 0, 1]]:
- Row-reduce the matrix (Gauss–Jordan elimination with partial pivoting)
- Count the non-zero rows: rank = 2
Nullity (columns − rank): 1 · Full rank: no
| col 1 | col 2 | col 3 | |
|---|---|---|---|
| row 1 | 1 | 0 | 1 |
| row 2 | 0 | 1 | 1 |
| row 3 | 0 | 0 | 0 |
GATE tip
Consistency questions are common in GATE: compare rank(A) with rank([A|b]) and the number of unknowns. For a 3 × 3 system, if det A ≠ 0 there is always exactly one solution, so you only need ranks when det A = 0.
More Linear Algebra Calculators
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LU Decomposition Calculator
A = LU, uᵢⱼ = aᵢⱼ − Σ lᵢₖuₖⱼ, lⱼᵢ = (aⱼᵢ − Σ lⱼₖuₖᵢ)/uᵢᵢ
Doolittle: L has 1s on the diagonal
Example: The matrix [[2, −1, −2], [−4, 6, 3], [−4, −2, 8]]: det A = product of U’s diagonal = 24.
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Gauss Elimination Calculator
Rᵢ → Rᵢ − (aᵢₖ/aₖₖ)·Rₖ
forward elimination to upper triangular form, then back substitution
Example: The system 2x₁ + x₂ − x₃ = 8, −3x₁ − x₂ + 2x₃ = −11, −2x₁ + x₂ + 2x₃ = −3: Solution: x1 = 2, x2 = 3, x3 = -1.
Frequently asked questions
How do I find the rank of a matrix quickly?
Row-reduce it and count the non-zero rows. A shortcut: if one row is a multiple of another, the rank drops by one; a non-zero determinant of an n × n matrix means full rank n.
What is the rank–nullity theorem?
For an m × n matrix, rank + nullity = n: the number of free variables in Ax = 0 equals the columns minus the rank.
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
Cite this page
APA
Scientific Converters. (2026, October 15). Matrix Rank Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/matrix-rank-calculator/
IEEE
Scientific Converters, "Matrix Rank Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/matrix-rank-calculator/