(a) Suppose that A is a 6 x 6 matrix with characteristic polynomial CA(A) = (1 + 2)3 (1 + 1)2 ( 1 - 1). Does there exist a set of three linearly independent vectors v1, V2, V3 in R such that Av1 = V1, Av2 = v2, and Av3 = V3? Justify your answer. (b) Let X and Y be square matrices of the same size. Suppose that A is an eigenvalue of X and / is an eigenvalue of Y. (i) By giving an example, verify that Ap need not be an eigenvalue of XY. (ii) Suppose that v is a common eigenvector of X and Y such that v is an eigenvector of X corresponding to the eigenvalue A of X and v is an eigenvector of Y corresponding to the eigenvalue / of Y. Show that Ap is an eigenvalue of XY