Most axiomatic system have an infinite number of theorems. I would say that a theory is "known" if there is an algorithm to decide which statements follow from the theory. An example is plane geometry, which has an algorithm due to Tarski. (The algorithm is pretty slow, though -- doubly exponential. Maybe a better criterion is a polynomial-time algorithm, though I don't know of a system that has such.)

Practically, questions about matrices are easy. (Matrices over the real and complex numbers can probably be reduced to the same Tarski algorithm, actually.) If you can reduce a question to a linear algebra question, you're probably going to solve it. There are open questions in linear algebra if you force the entries to be integers, for example, but odds are you don't have one of those.