It's basically duck-typing. If you can do arithmetic on it, it's a number. If you can do matrix operations, it's a matrix. If it satisfies the axioms of X, it's an X.
Matrix is not defined just by operations like a ring is, but also by structure - you have N independent axes in a specified space.
> If you can do numeric arithmetic on it, it's a number.
>you have N independent axes in a specified space.
Only if you have an operation that maps the matrices to that space.