Linear Algebra Similar Matrices Problem

[SuperZero]

Homework Statement

Find two 2 × 2 matrices A and B such that AB fails to be similar to
BA. Hint: it can be arranged that AB is zero but BA is not.

Homework Equations

N/A

The Attempt at a Solution

If AB similar to BA, AB = S^-1BAS for some S, so AB DNE S^-1BAS?
Or maybe det(AB) DNE det(BA)
I'm really not sure how to do this without guess and check, which I also tried and failed at

 

[Dick]

You aren't paying much attention to the hint. If AB=0 and BA is not zero, then they aren't similar. Look for a example of that.