Matrix Rank Calculator
Find the rank of a matrix — the number of linearly independent rows (or columns). Computed via Reduced Row Echelon Form (RREF).
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Notes
What is Matrix Rank?
The rank of a matrix is the maximum number of linearly independent rows or columns. It equals the number of pivot rows (non-zero rows) in the RREF of the matrix.
Example: Find the Rank via Row Reduction
Starting matrix A:
| 1 | 2 | 3 |
| 2 | 4 | 6 |
| 0 | 1 | 2 |
R2 → R2 − 2·R1 (eliminates second row):
| 1 | 2 | 3 |
| 0 | 0 | 0 |
| 0 | 1 | 2 |
Swap R2 ↔ R3, then RREF:
| 1 | 0 | -1 |
| 0 | 1 | 2 |
| 0 | 0 | 0 |
Two non-zero rows → rank(A) = 2.
A square n×n matrix has full rank (rank = n) if and only if its determinant is nonzero — i.e., it is invertible.
See also
- Matrix Calculations Guide — Matrix Rank — In-depth notes on Matrix Rank with worked examples
Frequently Asked Questions
What is the maximum possible rank of an m×n matrix?
The rank is at most min(m, n) — you cannot have more independent rows than you have rows, and similarly for columns.
What does rank 0 mean?
Rank 0 means every entry of the matrix is zero. The zero matrix has rank 0.
What is the nullity?
By the rank-nullity theorem, rank + nullity = n (number of columns). Nullity measures the dimension of the null space (solution space of Ax = 0).