It's not really necessary to argue. Cayley's theorem guarantees that every group is a subgroup of a symmetric group:
http://en.wikipedia.org/wiki/Cayley's_theorem
A symmetric group is then a subgroup of the general linear group in any field, where the general linear group GL(K, N) is the set of invertible NxN matrices with entries from a "field" (just think "numbers") and therefore any [finitely generated] group is isomorphic to a group of matrices under matrix multiplication; this is the underlying concept of representation theory.
If we back up a little, a group is a very general sort of algebraic structure; many important concepts have an underlying group structure, such as rotations, permutations, and any sort of reversible computation. This latter case implies that matrix multiplication is Turing complete; the simplest such set of matrices is generated by the Toffoli gate matrix. The relationship of groups to geometry is due to the underlying correspondence between the axioms of a group and those of geometrical transformations. A set of reversible geometric transformations includes an identity element -- do nothing -- and obeys associativity (sorta complicated, but it makes sense if you think about it) and inverse operations (by assumption): this makes it a group, and it can be represented by matrices. If we remove "reversible", we get exceptions -- like the cross product -- but these are usually related to groups (cross product -> quaternion algebra).
So matrix multiplication is actually really, really fundamental in a lot of mathematics. It's also a special case of tensor contraction, which could justify another tower post (but won't).
>And trust me, when you are writing numerical code, being able to read the mathematical formula clearly is essential, especially when you have a lot of formulas and need to figure out why your code is giving you numerical non-sense.
(spent three months chasing a bug where two commands were out of order)