# Matrix Multiplication Calculator

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

> The matrix multiplication calculator multiplies a matrix A (up to 3 × 3) by a matrix B (up to 3 × 3) when the inner dimensions match, showing how each entry is formed.

## Inputs

- First matrix size (rows × columns)
- A: a₁
- A: b₁
- A: c₁
- A: a₂
- A: b₂
- A: c₂
- A: a₃
- A: b₃
- A: c₃
- Second matrix size (rows × columns)
- B: x₁
- B: y₁
- B: z₁
- B: x₂
- B: y₂
- B: z₂
- B: x₃
- B: y₃
- B: z₃

## Results

- A × B
- Size of the product
- How each entry is computed

## Formula

```
cᵢⱼ = Σₖ aᵢₖ · bₖⱼ  (A is m × n, B is n × p, C is m × p)
```

## Explanation

Matrix multiplication composes linear transformations, chains Markov transitions, rotates and scales graphics, and is the core operation of neural networks.

### How to use the Matrix Multiplication Calculator

- **First matrix size (rows × columns)**
- **A: a₁**
- **A: b₁**
- **A: c₁**
- **A: a₂**
- **A: b₂**
- **A: c₂**
- **A: a₃**
- **A: b₃**
- **A: c₃**
- **Second matrix size (rows × columns)**
- **B: x₁**
- **B: y₁**
- **B: z₁**
- **B: x₂**
- **B: y₂**
- **B: z₂**
- **B: x₃**
- **B: y₃**
- **B: z₃**

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

### How it works

Each entry of the product is the dot product of a row of A with a column of B. The number of columns of A must equal the number of rows of B.

## Worked example

Inputs — First matrix size (rows × columns): 2 × 3; A: a₁: 4; A: b₁: 1; A: c₁: 2; A: a₂: 1; A: b₂: 3; A: c₂: 0; Second matrix size (rows × columns): 3 × 2; B: x₁: 1; B: y₁: 2; B: x₂: 0; B: y₂: 1; B: x₃: 2; B: y₃: 1.

Results — **A × B = ⎡ 8 11 ⎤
⎣ 1 5 ⎦**; Size of the product = 2×2; How each entry is computed = c₁₁ = 4·1 + 1·0 + 2·2 = 8
c₁₂ = 4·2 + 1·1 + 2·1 = 11
c₂₁ = 1·1 + 3·0 + 0·2 = 1
c₂₂ = 1·2 + 3·1 + 0·1 = 5.

## Key terms

- **Matrix product**: The composition of linear maps.

## FAQ

**When can two matrices be multiplied?**
When A has as many columns as B has rows.

**Is AB = BA?**
Generally not – matrix multiplication is not commutative.

**What is the size of the product?**
(rows of A) × (columns of B).

**What is the identity matrix?**
The square matrix with 1s on the diagonal; AI = IA = A.


## Sources

- [Wolfram MathWorld – Matrix multiplication](https://mathworld.wolfram.com/MatrixMultiplication.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/)