Also, remember that you don't have "matrices" in the usual sense, because matrices have one plug and one socket. What you have are two-index arrays of numbers enclosed in square brackets, which look like matrices because physicists don't like clarity. I mean, you don't really have a significantly better way to write out all of the components, but it's still a bad way to write them because it encourages this kind of mistake. But you can't just multiply your metric tensor by itself. Well, you can, but you had better end up with something that has four upper indices. Matrix multiplication is a form of contraction, and you need to both have upper indices and have lower indices for that.
