Matrix Multiplication

Matrix multiplication combines two matrices to produce a new matrix.

The Rule

For $C = AB$, each element is a dot product:

$$c_{ij} = \text{row } i \text{ of } A \cdot \text{column } j \text{ of } B$$

Dimensions

• $A$ is $m \times n$, $B$ is $n \times p$
• $C$ is $m \times p$
• The inner dimensions must match

Four Ways to Multiply

1. Row × Column — the standard dot product
2. Column picture — $AB$ = linear combination of $A$’s columns
3. Row picture — each row of $C$ = linear combination of $B$’s rows
4. Column × Row — sum of rank-1 matrices

Properties

• Not commutative: $AB \neq BA$ (usually)
• Associative: $(AB)C = A(BC)$
• Distributive: $A(B + C) = AB + AC$