Most of the time I see matrix multiplication presented and defined, as a seemingly arbitrary sequence of operations. For example, the textbook I'm currently reading for a linear algebra course defines the product $AB$ as the $(i, j)$ entry in the $1 \times 1$ matrix that is the product of the ith row of $A$ and the jth column of $B$. Properties of matrix multiplication are subsequently proven based upon this definition. The definition is clear, but why the matrix product is useful is not clear to me as a student. A different textbook I'm referencing defines the product $AB$ in terms of linear combinations.
The problem I have is doing matrix multiplication quickly by hand, particularly when the $A$ is $p \times 1$ and $B$ is $1 \times q$. I would like to know of how to look at or define matrix multiplication, in a manner which makes it easy (for the average student) to compute by hand, while being intuitive and consistent for use for later proofs.
wikipedia has a great article.