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(5) Suppose that T is a linear operator on a finite dimensional vector space V, that V1, ..., VK are T-invariant subspaces of V, and that V = VIOVO . . . O VK. Prove that det(T) = det(Tv, ) det(TV2 ) . . . det(TVk).(6) Let n E N. Find the characteristic polynomial of the matrix A E Mn,n (R) defined by 1 ... 1 1 . . . A = Hint: Use the fact that n(A) is the geometric multiplicity of the eigenvalue 0.(7) Let n E N. Find the characteristic polynomial of the matrix A E Mn,n (R) defined by 2 n n +1 n + 2 2n A = 2n + 1 2n + 2 3n . . n2 - n+1 n2 -n+2 .. n2 Hint: Use the hint from the previous exercise, and the fact that span {(1, 1, . .. 1), (1, 2, ..., n)t is a TA-invariant subspace of R"