> [!NOTE] Definiton (Matrix Multiplication) > Let $A,B\in \text{Mat}_{mn}(\mathbb{R})$ be [[Real Matrices|real matrices]] of order $m\times n$ and $n\times l$ respectively. Then product of $A$ and $B,$ denoted $AB,$ is given the matrix $C\in \text{Mat}_{ml}(\mathbb{R})$ whose $(i,j)$ entry is the [[Dot Product in Real n-Space|dot product]] of the $i$-th [[Matrix Row|row]] of $A$ and the $j$-th [[Matrix Column|column]] of $B,$ that is $c_{ij}=\sum_{k=1}^{n} a_{ik}b_{kj}$for all $(i,j)\in[1,2,\dots,m]\times[1,2,\dots,l].$ **Language**: Let $AB$ be a matrix product. We say that $A$ is post-multiplied by $B$ and that $B$ is pre-multiplied by $A.$