# Matrix Rank Calculator

URL: https://speedycalculator.com/calculators/math/matrix-rank/
Last updated: 2026-10-30

> The matrix rank calculator finds the rank of a matrix of up to 4 × 4 – the number of linearly independent rows or columns – with its reduced row echelon form.

## Inputs

- Matrix size (rows × columns)
- a₁
- b₁
- c₁
- d₁
- a₂
- b₂
- c₂
- d₂
- a₃
- b₃
- c₃
- d₃
- a₄
- b₄
- c₄
- d₄

## Results

- Rank
- Full rank?
- Reduced row echelon form

## Formula

```
rank(A) = number of pivots in rref(A) = dim Col(A) = dim Row(A)
```

## Explanation

The rank tells you how much independent information a matrix contains: whether a system of equations has a unique solution, whether vectors span a space and how many dimensions a transformation preserves.

### How to use the Matrix Rank Calculator

- Choose the **matrix size** – the matching grid of fields appears.
- Enter the entries **row by row** (a₁ b₁ c₁ is the first row, a₂ b₂ c₂ the second, and so on).
- Read the result; integer matrices give exact fractions.

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

The matrix is reduced to reduced row echelon form; the rank is the number of non-zero rows (pivots).

## Worked example

Inputs — Matrix size (rows × columns): 3 × 4; a₁: 1; b₁: 2; c₁: 3; d₁: 4; a₂: 2; b₂: 4; c₂: 6; d₂: 8; a₃: 0; b₃: 1; c₃: 1; d₃: 2.

Results — **Rank = 2**; Full rank? = Rank deficient; Reduced row echelon form = ⎡ 1 0 1 0 ⎤
⎢ 0 1 1 2 ⎥
⎣ 0 0 0 0 ⎦.

## Key terms

- **Rank**: The dimension of the row or column space.

## FAQ

**What is full rank?**
Rank equal to the smaller of the number of rows and columns.

**Can the rank exceed the number of rows?**
No – it is at most min(rows, columns).

**Does row reduction change the rank?**
No – elementary row operations preserve it.

**What is the rank of the zero matrix?**
0.


## Sources

- [Wolfram MathWorld – Matrix rank](https://mathworld.wolfram.com/MatrixRank.html)
- [MIT OpenCourseWare – 18.06 Linear Algebra (Gilbert Strang)](https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/)
- [Jim Hefferon – Linear Algebra (free textbook)](https://hefferon.net/linearalgebra/)