Problem 2 If true, write proof; If false, give a counter-example. (a) If A c Roxm and AA is invertible, then rank(A) = n. (b) For matrices A, BE Rox, if AB = BA = 0, then A = 0 or B = 0. (c) If Be Rox* is symmetric with strictly positive eigenvalues (i.e., > > 0 for i = 1, ..., n) then B is invertible. (d) For any Ac Rox" and any a > 0, A'A + al is always invertible. (e) For all A ( RX, if v1 ( R" ia an eigenvector of A associated with the eigenvalue An and if 12 6 R" is an eigenvector of A associated with eigenvalue )2 x X1, then v1 + v2 is an eigenvector of A associated with the eigenvalue >1 + 12