We will first prove that if then is a unit. This is simple, . Thus, are inverses of eachother. Say that is a unit. Since it means and so since the ring is commutative. Using the above result it means is a unit. But then is a unit.
So for the 2X2 Matirx would you take B to be the matrix (0 1, 0 0) as that squared = (0 0, 0 0)
This is confusing me slightly, I haven't seen the notation of a "unit" in this context before...What exactly would the conditions be I need to satisfy for A-B to not be a unit?