NAME: Student Number: Nov.16, 2015 MATHEMATICS 422 - Assignment 4 Due time for this assignment: 5:00pm, Nov.27, 2015. Group Discussions are encouraged. You may choose 5 questions. (Total: 10 marks) 1. Exercise 6 in Section 6.6. 2. Let p(t), q(t) C[t] be relatively prime, A Mn (C). Show that rank(p(A))+rank(q(A)) n. 3. Let V be a nite-dimensional vector space over a eld F , and f L(V ). Show that V = ker(f ) Im(F ) i Im(f 2 ) = Im(f ). 4. Let V be a nite-dimensional vector space over a eld F , and f L(V ). Show that there is an invertible g L(V ) such that gf gf = gf . 5. Let A Mn (C) such that there exists a positive integer k for which kAk+1 = (k + 1)Ak . Prove that the matrix A In is invertible and nd its inverse. 6. Let A Mn (C) with rank(In + A)+rank(In A) = n. Show that A2 = In . (Hints: Use Jordan canonical form of A.)