For all questions below assume that T V V is a linear operat

For all questions below, assume that T V V is a linear operator on a finite-dimensional vector space

Solution

As S has 4 eigenvalues (with possible repetitions) with exaclty 3,4, 7 as the eigenvalues, it follows that exatly one of them is repeated , say 3.

By Cayley Hamilton theorem

                                                  (S-3)² (S-4) (S-7) =0

which implies                             (S-3)² (S-4)² (S-7)² = (S-4)(S-7) [ (S-3)² (S-4) (S-7)]

                                                                                   = (S-4)(S-7).o

                                                                                     =0

as required.

Hence proved