2. [H] Suppose A and B are matrices such that both AB and BA are defined. a) Show that AB and BA are both square matrices. b) If AB = BA, show that 4 and B are both square and of the same size. ) If A and B are square matrices such that AB = BA, show that (A B)(A+ B)= A? - B~ d) Find two 2 x 2 matrices A, B for which (A B)(A + B) # A% B2, e) Prove that (A + B)? = A* + B* + 2AB if and only if AB = BA. Let A and B be matrices of the same size. By considering the general entries [A];;. [Bli;, [A + B)i; and [B + A];j, prove the commutative laws of addition, ie. A+ B =B+ A Suppose A is a scalar and A. B M,,,. Prove that AM(A+ B) = AA + AB. Let A and B be two matrices such that AF is defined. By considering the general entry in both sides of the equation, show that A(AB) = AAB where A is any real number