# Matrix Calculator

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

> The matrix calculator performs the common operations on 2 × 2 and 3 × 3 matrices: determinant, inverse, transpose, trace, rank, addition, subtraction, multiplication, scalar multiplication and powers.

## Inputs

- Size
- Operation
- A: a₁
- A: b₁
- A: c₁
- A: a₂
- A: b₂
- A: c₂
- A: a₃
- A: b₃
- A: c₃
- B: x₁
- B: y₁
- B: z₁
- B: x₂
- B: y₂
- B: z₂
- B: x₃
- B: y₃
- B: z₃
- Number k (or power n)

## Results

- Result
- Details

## Formula

```
det, A⁻¹, Aᵀ, tr(A), rank, A ± B, A·B, k·A, Aⁿ
```

## Explanation

Matrices organise numbers in rows and columns and describe linear transformations, systems of equations, networks and images. This calculator is a quick all-in-one tool for the standard operations.

### How to use the Matrix 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

Pick the operation; matrix B or the number k appear only when they are needed. Integer inputs give exact fraction results.

## Worked example

Inputs — Size: 2 × 2; Operation: Determinant; A: a₁: 2; A: b₁: 1; A: a₂: 1; A: b₂: 4.

Results — **Result = 7**; Details = |A| = a₁·b₂ − b₁·a₂ = (2)(4) − (1)(1) = 7.

## Key terms

- **Matrix**: A rectangular array of numbers.

## FAQ

**How do I multiply two matrices?**
Each entry is a row of A dotted with a column of B.

**Can I add matrices of different sizes?**
No – addition needs the same dimensions.

**What is the trace?**
The sum of the diagonal entries.

**Why might the inverse not exist?**
The determinant is zero.


## Sources

- [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/)