Consider the matrix 0 1 1 A = 101 0 (i) Use the characteristic polynomial f(x) := det(x] - A) to show that A has only two distinct eigenvalues: 1, 12 = 13, say. Hence confirm that f (a) = (2 ->1)(2-12)2. (ii) Setting p(x) = (x - 1)(x - 12), use matrix algebra to explicitly show that p( A) = 0. (iii) Consider the subset Eig 12) C R3 of eigenvectors of A with eigenvalue 12, i.e.: Eig( A, 12) := {U ER3 : AU = 127}. Show that U E Eig( A, 12) whenever U = (A - All)u for some u E R3. (iv) Show that Eig A, 12) is a two-dimensional subspace of R3