Product of Real Matrices

Let $m,n,p\in \mathbb{N}$. Let $A=(A_{\alpha i})$ be an $m\times n$ matrix with real entries, and let $B=(B_{i\beta})$ be an $n\times p$ matrix with real entries. The product matrix $AB$ is the $m\times p$ matrix whose $(\alpha,\beta)$ entry is defined by

for every $\alpha\in\{1,\dots,m\}$ and every $\beta\in\{1,\dots,p\}$.