(i) Let be an -dimensional vector space over and let be an endomorphism. Suppose that the characteristic polynomial of is , where the are distinct and for every .
Describe all possibilities for the minimal polynomial and prove that there are no further ones.
(ii) Give an example of a matrix for which both the characteristic and the minimal polynomial are .
(iii) Give an example of two matrices with the same rank and the same minimal and characteristic polynomials such that there is no invertible matrix with .