I'm working from the perspective of computer graphics and I'd like to understand how we came to use matrices to express transformations in $R^2$ and $R^3$ that represent things like rotation, scale, and skew and the composition of those transformations.
My understanding is that matrices first appeared as mere abbreviations for systems of linear equations. And that later Gauss showed that they could represent linear transformations as well. So I can see that at that point we could define matrices to represent arbitrary transformations. But it's not clear why this was important. Was it simply the convenience and compact representation?
And second, were matrix multiplication defined in such a way as to allow us to compose these transformations or was it defined already and it just happened that that allowed the transforms to be composed? That seems like magic.
If the matrix operations came later, then I also wonder what addition was intended to represent? I don't know of any useful geometric meaning.