Definition:Matrix Product (Conventional)

Definition
Let $\left({R, +, \circ}\right)$ be a ring.

Let $\mathbf A = \left[{a}\right]_{m n}$ be an $m \times n$ matrix over $R$.

Let $\mathbf B = \left[{b}\right]_{n p}$ 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 $\mathbf A \mathbf B = \mathbf C = \left[{c}\right]_{m p}$.

Then:
 * $\displaystyle \forall i \in \left[{1 \, . \, . \, m}\right], j \in \left[{1 \, . \, . \, p}\right]: c_{i j} = \sum_{k=1}^n a_{i k} \circ b_{k j}$

Thus $\left[{c}\right]_{m p}$ is the $m \times p$ matrix where each element $c_{i j}$ is built by forming the product of each element in the $i$'th row of $\mathbf A$ with the corresponding element 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.

Linguistic Note
Some older sources use the term matric multiplication; strictly speaking it is more correct, as matric is the adjective formed from the noun matrix, but it is a little old-fashioned and is rarely found nowadays.

Historical Note
This mathematical process was first introduced by Jacques Philippe Marie Binet.

Beware
We do not use $\mathbf A \times \mathbf B$ or $\mathbf A \cdot \mathbf B$ in this context, because they have specialised meanings.