Matrix Multiplication Calculator
Multiply two matrices of compatible dimensions with full row-by-column working.
Inputs
A × B
1 2 3 4
Result Dimensions
2×2
Step by step
Matrix A
= 1 2 3 4
Matrix B
= 1 0 0 1
A × B
= 1 2 3 4
Note
(A×B)ᵢⱼ = sum of row i of A times column j of B
How it works
Matrix multiplication computes (AB)ᵢⱼ as the dot product of row i of A with column j of B. The number of columns in A must equal the number of rows in B. The result is an m×n matrix when A is m×p and B is p×n. Matrix multiplication is not commutative in general.
Formula
Matrix Multiplication
(AB)ᵢⱼ = Σₖ Aᵢₖ · Bₖⱼ
- A
- Left matrix (m×p)
- B
- Right matrix (p×n)
Frequently Asked Questions
Is matrix multiplication commutative?
No. In general AB ≠ BA. The dimensions may not even permit the reverse product.
What happens when I multiply by the identity matrix?
Multiplying any matrix A by the identity I gives A back: AI = IA = A.