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
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