Multiplication

Definition: The product of an $m \times n$ matrix $A$ and an $n \times p$ matrix $B$ is the $m \times p$ matrix $C = AB$ defined by:

$$c_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}$$

The entry $c_{ij}$ is the dot product of row $i$ of $A$ with column $j$ of $B$.

Example:
$$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}$$

Properties:

• Associative: $(AB)C = A(BC)$
• Distributive: $A(B + C) = AB + AC$
• Identity: $AI = IA = A$
• NOT commutative: In general, $AB \neq BA$

Important notes:

• Requires the number of columns of $A$ to equal the number of rows of $B$
• $(AB)^T = B^T A^T$
• $\det(AB) = \det(A)\det(B)$ for square matrices