# Matrix Determinant Calculator

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

> The matrix determinant calculator computes the determinant of a 2 × 2, 3 × 3 or 4 × 4 matrix, with the cofactor expansion written out.

## Inputs

- Matrix size
- a₁
- b₁
- c₁
- d₁
- a₂
- b₂
- c₂
- d₂
- a₃
- b₃
- c₃
- d₃
- a₄
- b₄
- c₄
- d₄

## Results

- Determinant |A|
- Expansion
- Matrix

## Formula

```
2 × 2: |A| = a₁b₂ − b₁a₂
n × n: |A| = Σⱼ (−1)^(1+j) a₁ⱼ M₁ⱼ
```

## Explanation

The determinant tells you whether a matrix is invertible (non-zero), by how much it scales areas and volumes, and whether it flips orientation (negative). It appears in Cramer's rule, eigenvalues and changes of variables in integrals.

### How to use the Matrix Determinant 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

For 2 × 2 the cross-product formula is used; for larger matrices the calculator uses elimination for the value and shows the first-row cofactor expansion.

## Worked example

Inputs — Matrix size: 3 × 3; a₁: 4; b₁: 1; c₁: 2; a₂: 1; b₂: 3; c₂: 0; a₃: 2; b₃: 0; c₃: 5.

Results — **Determinant |A| = 43**; Expansion = Expand along the first row (cofactor expansion):
|A| = 4 × 15 − 1 × 5 + 2 × −6
= 60 − 5 − 12 = 43; Matrix = ⎡ 4 1 2 ⎤
⎢ 1 3 0 ⎥
⎣ 2 0 5 ⎦.

## Key terms

- **Determinant**: A scalar summarising a square matrix.

## FAQ

**What does a zero determinant mean?**
The matrix is singular – its columns are dependent and it has no inverse.

**What is the geometric meaning?**
The factor by which the transformation scales areas (2D) or volumes (3D).

**What happens if I swap two rows?**
The determinant changes sign.

**Is det(AB) = det(A)·det(B)?**
Yes.


## Sources

- [Wolfram MathWorld – Determinant](https://mathworld.wolfram.com/Determinant.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/)