While certain “natural” properties of multiplication do not hold, many more do. In the next subsection, we will state and prove the relevant theorems. But first, we need a theorem that provides an alternate means of multiplying two matrices. In many texts, this would be given as the definition of matrix multiplication. We prefer to turn it around and have the following formula as a consequence of our definition. It will prove useful for proofs of matrix equality, where we need to examine products of matrices, entry-by-entry.
Suppose $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix. Then for $1\leq i\leq m$, $1\leq j\leq p$, the individual entries of $AB$ are given by \begin{align*} \matrixentry{AB}{ij} &= \matrixentry{A}{i1}\matrixentry{B}{1j}+ \matrixentry{A}{i2}\matrixentry{B}{2j}+ \matrixentry{A}{i3}\matrixentry{B}{3j}+ \cdots+ \matrixentry{A}{in}\matrixentry{B}{nj}\\ &= \sum_{k=1}^{n}\matrixentry{A}{ik}\matrixentry{B}{kj} \end{align*}
Theorem EMP is the way many people compute matrix products by hand. It will also be very useful for the theorems we are going to prove shortly. However, the definition (Definition MM) is frequently the most useful for its connections with deeper ideas like the null space and the upcoming column space.