Please show me step by step those questions, I will rate them immediately! Thank You

(1) Give an example of an invertible linear operator T on a vector space V, and a T invariant subspace W E V with the property that W is not T'l-invariant. \f(3) Suppose that A E Mnn (C) has characteristic polynomial n f (2) = i=0 and that f(0) = 1. Prove that A is invertible and that n-1 A- 1 = aitIA? i=0(4) Let ao, . .., an-1 E C and let A E Mn,n (C) be defined by 10 0 0 ... O do 10 0 . . a1 A = ] 01 0 a2 . . . . . . . O . .. 1 an-1/ Prove that the characteristic polynomial of A is n-1 f (a) = (-1)n-1 i=0