I've seen "proofs" of this before but they all just assume we already know that matrix multiplication is associative, matrices with nonzero determinant are invertible, etc. and I do know this insofar that I've been told it since high school, but I don't think I've actually ever seen a proof that matrix multiplication is well defined, associative, etc. Does anyone know of any more rigorous proofs out there? I've googled a lot but haven't found anything. It's not exactly the sort of thing I'd want to spend time on proving rigorously myself since I already know these things to be true, but I would like to see a proof.