\begin{definition}[Definition:Matrix Product (Conventional)]
Let $\struct {R, +, \circ}$ be a ring.
Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over $R$.
Let $\mathbf B = \sqbrk b_{n p}$ be an $n \times p$ matrix over $R$.
Then the '''matrix product of $\mathbf A$ and $\mathbf B$''' is written $\mathbf A \mathbf B$ and is defined as follows.
Let $\mathbf A \mathbf B = \mathbf C = \sqbrk c_{m p}$.
Then:
:$\ds \forall i \in \closedint 1 m, j \in \closedint 1 p: c_{i j} = \sum_{k \mathop = 1}^n a_{i k} \circ b_{k j}$
Thus $\sqbrk c_{m p}$ is the $m \times p$ matrix where each entry $c_{i j}$ is built by forming the (ring) product of each entry in the $i$'th row of $\mathbf A$ with the corresponding entry in the $j$'th column of $\mathbf B$ and adding up all those products.
This operation is called '''matrix multiplication''', and $\mathbf C$ is the '''matrix product''' of $\mathbf A$ with $\mathbf B$.
It follows that '''matrix multiplication''' is defined whenever the first matrix has the same number of columns as the second matrix has rows.
\end{definition}