✕ Matrix Multiplication Calculator
Matrix Multiplication Calculator
Multiply two matrices A × B with full element-by-element computation shown. Auto-checks dimension compatibility.
Matrix Multiplication A × B
Enter matrices A and B with compatible dimensions
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ℹ For A (m×n) × B (n×p): the number of columns in A must equal the number of rows in B.
Matrix A
×
Matrix B
Quick Examples
Matrix Multiplication Rules
The product of A (m×n) and B (n×p) is matrix C (m×p) where:
$$C_{ij} = \sum_{k=1}^{n} A_{ik} \cdot B_{kj}$$
Each element of C is the dot product of row i of A with column j of B.
Key Properties
- Not commutative: AB ≠ BA in general
- Associative: (AB)C = A(BC)
- Distributive: A(B+C) = AB + AC
- Dimension rule: A(m×n) × B(n×p) = C(m×p)
- det(AB) = det(A) × det(B) for square matrices