a) Prove that if v is an eigenvector of a matrix A, then for any nonzero scalar c, cv is also an eigenvector of A. b) Prove that if v is an eigenvector of a matrix A, then there is a unique scalar lambda such that Av = lambda v. c) Prove that a square matrix is invertible if and only if 0 is not an eigenvalue. d) Prove that if lambda is an eigenvalue of an invertible matrix A, then lambda notequalto 0 and 1/lambda is an eigenvalue of A^-1. e) Prove that if lambda is an eigenvalue of a matrix A, then loambda^2 is an eigenvalue of A^2.