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.
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