I think it is much more intuitive and powerful to make the reverse reasoning. Linear algebra studies vectors belonging to some vector space and the linear transformations between those spaces. You can then realize that abstract concept with, for instance, ordered 3-tuples of real numbers, in which case 3x3 matrices represent linear mapping between those R^3 vector spaces, in which case we can use it to do physics in a 3D world. But it's misguided to say matrices come "before" abstract linear algebra.