What does the Matrix Multiplication calculator do?

It computes matrix multiplication directly in your browser and shows results immediately.

Is MatrixDojo free to use?

Yes. MatrixDojo is free to use for matrix calculations with no signup required.

Matrix Multiplication Calculator

Enter two matrices and get the product instantly. Supports fractions, variables, and shows the full working.

Matrix A

Rows

Cols

Supports decimals (0.75), variables (x, alpha), and expressions (2x + 1).

Matrix B

Rows

Cols

Supports decimals (0.75), variables (x, alpha), and expressions (2x + 1).

Result: A × B

2 × 2

-10

2 × 2

Step-by-step solution

Step 1 — Setup

A is 2×3, B is 3×2. Result C = A·B is 2×2.
Each entry C[i][j] = sum over k of A[i][k] · B[k][j].

Step 2 — C[1][1]
```
(1)·(2) + (0)·(0) + (3)·(-4) = -10
```
Step 3 — C[1][2]
```
(1)·(-1) + (0)·(3) + (3)·(1) = 2
```
Step 4 — C[2][1]
```
(0)·(2) + (2)·(0) + (-1)·(-4) = 4
```
Step 5 — C[2][2]
```
(0)·(-1) + (2)·(3) + (-1)·(1) = 5
```

How matrix multiplication works

To multiply matrix A (size m×n) by matrix B (size n×p), the number of columns in A must equal the number of rows in B. Each entry of the resulting m×p matrix is the dot product of the corresponding row of A and column of B: (AB)ᵢⱼ = Σₖ AᵢₖBₖⱼ.

This calculator supports decimals like 0.75, variables like x or alpha, and full expressions like 2x + 1. Everything runs in your browser — no data is uploaded.

Order matters in general: A×B is usually different from B×A, and one may be undefined.
Result shape is easy to predict before computing: (m×n)·(n×p) always returns m×p.
Associative: (AB)C = A(BC), so chained products can be regrouped when dimensions match.
Distributive: A(B + C) = AB + AC and (A + B)C = AC + BC.
Identity and zero rules: AI = IA = A, and A0 = 0, 0A = 0.