To multiply the $m$ x $n$ matrix $A$ by the ${} n$ x ${} p$ matrix $B$, each element of the resulting $m$ x ${} p$ matrix $C$ is the dot product of the corresponding rows of $A$ and the columns of $C$. This can be approached in a few different ways: ![[Pasted image 20251229172211.png|400]] Basic identities: $A(BC) = (AB)C$ $A(B+C)=AB+AC$ $(A+B)C=AC+BC$ $I_{m}A = A + AI_{m}$