The [[Matrix Product]] is a notion of 'multiplying' two [[Matrix|Matrices]]. The [[Matrix Product]] is a non [[Commutative]], [[Associative]] [[Binary Operation]] between a matrix $A\in M_{m\times n}$ and another $B\in M_{n\times p}$ produces another [[Matrix]] $AB\in M_{m\times n}$ $ \huge \begin{align} \forall A &\in M_{n\times m}, B \in M_{m \times r} \\ AB &\in M_{n\times r} \end{align} $ $\huge \begin{align} \begin{split} AB &= \mat{A\vec B_{1} A\vec B_{2}, \dots,A\vec B_{n} } \\ \pa{AB}_{i,j} &= \sum_{k=1}^{m} a_{i,k}b_{k,j} \end{split} \end{align}$ Matrix products are *not* [[Commutative]], $\exists A,B \ba{ AB \ne BA}$. $\huge \begin{align} \forall k&\in R \\ (kA)B &=k(AB)=kAB \\ \end{align} $