Ignore post, I found a counterexample to (2). I'm studying for an upcoming exam, and I'm a bit confused about how to go about proving or disproving the statement (2). 1.) Products of diagonalizable matrices are never diagonalizable. I figured false and my counterexample is really just the square of the identity matrix. Since I is diagonalizable and I^2=I. 2.) Productions of diagonalizable matrices are always diagonalizable. I'm not too sure if it's true to begin with. Trying to construct a counterexample for 2x2 matrices hasn't been successful yet. Any hints much appreciated.