Any linear map between finite-dimensional vector spaces may be represented by a matrix, and conversely. Matrix-matrix multiplication corresponds to map composition, and matrix-vector multiplication corresponds to map application.
This provides a nice way of understanding matrices and the origin of the rules for their calculation. Having attained this perspective is a huge boon to me, because I've always resented that matrix math is basically just taught as a collection of arbitrary manipulations of symbols (multiply the entries! now add them! don't ask why!).
But is this understanding always meaningful?
For in general, just because you've found an interpretation for a mathematical concept doesn't mean it will always apply. For instance, the trig functions can be thought of as simply being the coordinates on a unit circle, but this fails to explain why they show up as solutions to differential equations having nothing to do with circles (or indeed geometry) - to explain this we require a more abstract understanding of the trig functions.
Thus, in all non-trivial uses of matrices in pure and applied mathematics, can we meaningfully say that the matrices involved represent, at least implicitly, linear maps? For instance, we can think of solving a system of equations as simply trying to find a vector which, when a given map is applied to it, yields a specific image.
By non-trivial, I mean that the particular use of matrices involves, at minimum, matrix multiplication and/or determinants or other elementary aspects of the calculus of matrices. For instance, you could call a spreadsheet a matrix, but unless you intend to do anything matrix-like with it apart from draw it as a grid, you may as well just call it a set of numbers.
Note: this question may be soft, but I feel it can be given a clean answer. Either you can come up with a situation where the linear-map interpretation doesn't apply, or you can't.