I think the key here is that matrices are a "higher-kinded" data structure. That is, they're parameterized not only by the shape (number of rows and columns) bu

I think the key here is that matrices are a "higher-kinded" data structure. That is, they're parameterized not only by the shape (number of rows and columns) but also by the type of the elements themselves. Furthermore, the elements of a matrix need not be plain numbers (such as plain integers or floats) but can themselves be elements of any sort of ring, such as polynomials or even elements of a quotient ring. Now, you may think a matrix whose elements are polynomials with coefficients that are integers modulo some prime p with the polynomial ring itself quotiented by some irreducible polynomial might be too bizarre and esoteric to be useful but in fact these structures do see a lot of use in cryptography.