Matrix product

Set

 context $R$ … ring context $m,n,k\in \mathbb N$
 definiendum $*: \mathrm{Matrix}(m,n,R)\times \mathrm{Matrix}(n,k,R)\to \mathrm{Matrix}(m,k,R)$
 postulate $(A*B)_{ij}=\sum_{l=1}^m A_{il}\cdot B_{lj}$

Discussion

For square matrices, the matrix product is associative. And also for general matrices, we still have $(A*B)*'C=A*''(B*'''C)$, where the four binary functions are the matrix products for the suitable dimensions.