Matrix Multiplication

Combining linear transformations.

×How it Works

C = A × B

Element c_ij is the dot product of Row i of A and Column j of B.

Dimensionality Rule

If A is size (m × n) and B is size (n × p), then C will be size (m × p).
The inner dimensions (n) must match!

Properties

⚠Not Commutative: In general, AB ≠ BA. Order matters!
✓Associative: (AB)C = A(BC).
✓Distributive: A(B + C) = AB + AC.
✓Identity: AI = IA = A.

Geometric Meaning

Multiplying matrices corresponds to composing linear transformations. If matrix B rotates a vector, and matrix A scales it, then AB represents the combined operation: rotate then scale.