Matrix Multiplication Calculator
Multiply two matrices together.
Select Matrix Sizes
How Matrix Multiplication Works
Matrix multiplication is defined for matrices where the number of columns in the first matrix equals the number of rows in the second matrix.
For matrices A (m×n) and B (n×p), their product C = AB is an m×p matrix where:
\[ c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj} \]
Example for 2×2 matrices:
\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} ae+bg & af+bh \\ ce+dg & cf+dh \end{pmatrix} \]
Properties:
- Not commutative: AB ≠ BA in general
- Associative: (AB)C = A(BC)
- Distributive: A(B+C) = AB + AC
- Identity: AI = IA = A
- Zero: A0 = 0A = 0
Applications:
- Linear transformations
- Computer graphics
- Quantum mechanics
- Neural networks
- Economics and finance
Visual Interpretation:
- Each element in the result is a dot product
- Rows of A are multiplied by columns of B
- Result dimensions are (rows of A) × (columns of B)