37. Matrix Multiplication

For matrices () and (), the product is the matrix with entries:

Each entry is the dot product of row of with column of .

Compatibility: inner dimensions must match — ‘s columns must equal ‘s rows.

37.1. Properties

• Associative:
• Distributive: and
• Not commutative: in general
• Identity:
• Transpose:
• Inverse (when defined):
• Determinant:
• Zero product: does not imply or

37.2. Geometric meaning

If and represent linear transformations, then represents the composition — apply first, then (see Composition of Linear Transformations).

37.3. Special cases

• where is a single column → matrix–vector product
• where both are vectors → dot product (1×1 result)
• Outer product: column vector × row vector = rank-1 matrix