Let A be m ( n and B be n ( k. (a) Prove that rank(A B)

Question:

Let A be m ( n and B be n ( k.
(a) Prove that rank(A B) ( min {rank A, rank B}.
(b) Find A and B such that rank(A B) < min{rank A, rank B).
(c) If k = n and B is nonsingular, prove that rank(A B) = rank A.
(d) If m = n and A is nonsingular, prove that rank(A B) = rank B.
(e) For nonsingular matrices P and Q, what is rank(P A Q)?