Let $A$ be a $n \times m$ matrix and $B$ a $m\times p$ matrix, the product $AB$ is defined as a matrix of size $n\times p$ such that $(AB)_i,_j = \sum\limits_{k=1}^mA_i,_kB_i,_j$.

I was wondering for what good reason did mathematicians define it like this. Does somebody know? Thanks in advance.