Product of Diagonalizable matrices

[Klungo]

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.

 

[HallsofIvy]

A matrix, A, on Rⁿ, is "diagonalizable" if and only if there exist a basis for Rⁿ consisting of eigenvectors of A. Can you use that?