For matrices $A$ (size $m \times p$) and $B$ (size $p \times n$), the product $C = AB$ is defined as:

$ C_{ij} = \sum_{k=1}^{p} A_{ik}B_{kj} $

Key Properties:

• Compatibility: Number of columns in $A$ must equal number of rows in $B$
• Result size: If $A$ is $m \times p$ and $B$ is $p \times n$, then $AB$ is $m \times n$
• Non-commutative: Generally $AB \neq BA$
• Associative: $(AB)C = A(BC)$
• Distributive: $A(B + C) = AB + AC$

Applications:

• Composing linear transformations
• Solving systems of equations
• Graph theory (adjacency matrices)
• Quantum mechanics (operator composition)