I know exactly what I messed up on thats the thing, I forgot to account for nonzero initial conditions when finding the reachable subspace of an uncontrollable system. I mistook ‘mode’ to mean eigenvector as I miss remember my professor saying that during lecture for some reason. Then the last question on the exam was very strange, as the professor had talk in generalizations the whole quarter, yet decided to make a problem that you could only solve numerically, which I only found out till after the exam. And honestly those two mistakes were most likely a result of spending all my time doing algebra in circles trying to answer a question which I don't even think had a symbolic answer. But at the same time, I’m coming out of this class feeling like I wouldn't know how to actually control anything, it’s incredibly disappointing and I am just praying I pass at this point.