Math
Matrix Rank Calculator
Enter a matrix to find its rank, the number of linearly independent rows (or columns).
The calculator row-reduces to echelon form with exact fractions, shows each operation, counts the pivots, and reports the nullity, pivot columns, and whether the matrix has full rank.
One row per line. Separate entries with spaces or commas. Integers, decimals, and fractions like 1/3 all work. Up to 8 x 8.
Try an example
Result
Rank
—
- Maximum possible rank
- —
- Nullity
- —
- Full rank?
- —
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What the rank of a matrix means
The rank is the number of linearly independent rows, which always equals the number of linearly independent columns. It is the dimension of the column space (and of the row space), and it tells you how many independent equations a linear system really contains.
How to find the rank
- Row-reduce the A rectangular grid of numbers arranged in rows and columns. to row echelon form using swaps, scaling, and adding multiples of rows.
- Count the nonzero rows. Each one contains a pivot.
- That count is the rank. It cannot exceed min(rows, columns).
How to read the answer
| Result | Meaning |
|---|---|
| rank = min(rows, cols) | Full rank; for a square matrix, invertible with det ≠ 0 |
| rank < min(rows, cols) | Some rows/columns are combinations of others; square matrices are singular |
| rank = 0 | The zero matrix |
| nullity > 0 | The null space contains nonzero vectors; Ax = 0 has non-trivial solutions |
Common mistakes
- Counting nonzero rows of the original matrix instead of the echelon form. A matrix can have three nonzero rows and rank 1.
- Assuming a rectangular matrix has rank equal to its number of rows. The rank is at most the smaller dimension.
- Using a rounded decimal reduction and reading a tiny nonzero pivot as real. This calculator uses exact fractions, so a zero row is truly zero.
How to type a matrix into this calculator
- Put each row on its own line. Separate the entries in a row with spaces or commas: 2 1 1 or 2, 1, 1.
- Fractions such as 1/3 and decimals such as 0.25 are accepted and kept exact, so 1/3 stays 1/3 instead of becoming 0.3333.
- Every row must have the same number of entries. A row with a missing entry produces a clear error instead of a wrong answer.
- The largest size supported is 8 x 8. Brackets and semicolons are ignored, so [1 2; 3 4] works too.
Worked examples
Rank-deficient 3 x 3
Row 2 is twice row 1, so rank = 2
Rank
2
Full rank
Identity-like 3 x 3 has rank 3
Rank
3
Rectangular 2 x 4
Rank is at most 2
Rank
2
Rank 1
Every row is a multiple of (1, -1, 2)
Rank
1
Zero matrix
Rank 0, nullity equals the number of columns
Rank
0
Frequently asked questions
What is the rank of a matrix in simple terms?+
It is how many genuinely different rows (or columns) the matrix has after removing any that are combinations of the others. A rank-1 matrix is essentially one row repeated with different scalings.
Can the rank be larger than the number of rows?+
No. The rank is at most the smaller of the number of rows and the number of columns.
Is the row rank always equal to the column rank?+
Yes. This is a theorem of linear algebra, which is why the calculator shows rank(Aᵀ) equal to rank(A).
What is the nullity?+
The dimension of the null space, i.e. the number of free variables in Ax = 0. By the rank-nullity theorem it equals columns minus rank.
How does rank relate to the determinant?+
A square n x n matrix has rank n exactly when its determinant is nonzero. Rank below n means det = 0 and the matrix is singular.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026