When I took linear algebra class in my freshman year, the multiplication operation for matrices was defined without any apparent motivation. Given an $m$-times-$n$ matrix $A$ and an $n$-times-$p$ matrix $B$, define $(AB)_{ij} := \sum_{k=1}^n A_{ik}B_{kj}$.
I recently read a book which treats linear algebra more abstractly, introducing the notion of a matrix after it introduces linear maps. (The book is Linear Algebra Done Right by Sheldon Axler.)
Let $V$ and $U$ be vector spaces with bases $\{v_1,\ldots,v_n\}$ and $\{u_1,\ldots,u_m\}$. Notice that once we know how a linear map $T:U\to V$ acts on the basis elements of $U$, there is a unique way to extend it to the map on the whole space. The author then goes on to argue that we can thus associate with each linear map a matrix of that map. They are a very compact way of specifying linear maps relative to a given pair of bases.
We can define operations of addition and scalar multiplication on matrices so that they coincide with how these two operations act on the set of linear maps: $M(S+T)=M(S)+M(T)$ and $M(cT)=cM(T)$ for a scalar $c$ and linear maps $S,T:U\to V$.
What I like the most about the book is the next step.
Let $M(T)$ denote the matrix of the linear map $T:U\to V$ (relative, say, to standard basis). We know how to compose two linear maps $S:U\to V$ and $T:V\to W$ to get a linear map $ST:U\to W$.
What is more natural than to define matrix multiplication so that the equation $M(ST)=M(S)M(T)$ is satisfied? And we get the non-obvious rule for matrix multiplication. Despite not being simple it feels natural. I like it. I know where it came from. It makes sense.
Was this the original motivation for matrix multiplication? Who introduced it and when? (If not, when did people realize that is satisfies this rule?)
Edit (July 19th 2015). There is a substantial overlap with the question Why, historically, do we multiply matrices as we do? but that answer does not have a satisfactory answer. In particular, the URL in the accepted answer is dead.
