Q2. The commutator of A, B E Mnxn (F) is defined by [A, B] = AB — BA. Note that A and B commute if and only if their commutator is

zero. (a) Let B = Mnxn (F) be fixed. Define CB: Mnxn (F)→ Mnxn (F) by CB(A) = [A, B]. Prove that CB is a linear map. - J Determine rank(CB) and nullity (CB), where CB is as in (b) Let n 2 and B = part (a). (c) Again, let n = 2. Show that there is no B € M2x2(F) for which the map Cg is an isomorphism. [Hint: Compute [A, B] for a few matrices and then compute tr([A, B]). Do you notice anything peculiar?] Page 1 of 2