Suppose that A is an m × n matrix and that B is an n x p matrix. (a) im(A) related?) Show that rank(AB) < rank(A). (Hint: How are the subspaces im(AB) and (b) to part (a) to show that the 3 x 3 matrix AB cannot be invertible. Suppose that A is a 3 × 2 matrix and B is a 2 × 3 matrix. Use your answer (c) Show that dim ker(B) < dim ker(AB). In addition to what was shown in part (a) of this problem, it is also true (d) that rank(AB) < rank(B). Use the result of part (c), together with the rank-nullity theorem, to show this fact.

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