Math Help - diagonalizability of B if AB=BA

If A is diagonalizable and AB=BA, then is B diagonalizable? If so, how would I go about proving this? I think that this is true, but I am not positive.

Also, is there a real 2x2 matrix A satisfying A squared=-I? I can't think of one off of the top of my head.

Thanks!

If A is diagonalizable, then there exists an invertible matrix P and a diagonal matrix D such that . Specifically, A is also invertible since but D is diagonal and therefore obviously .

Now, what does this tell you about the original equation?

For the second, consider the 90 degree rotation matrix