Let a and b be real numbers satisfying a? - a - 1 =0 and b2 - 36 - 2 = 0, and let V = ab (a) Find a 4 x 4 matrix A with integer entries such that Av = av, and find a 4 x 4 matrix B with integer entries such that Bv = bv. Hint: Note that a2 = 1 +a and b2 = 2+3b. Do not try to use the quadratic formula to express a and b in terms of square roots, because that formula will not help here. (b) Show that (A+ B)v = (a+ b)v and that ABv = bav = abv. Take care with the order of multiplication when treating ABv. (c) Deduce from part (b) that there are polynomials f(x) = x* + cax3 + car'+ qir+ co g(x) = ' + dax + dax' + dix + do such that f(a + b) = 0 and g(ab) = 0, where co, . . ., Ca, do, . . ., d3 are integers. You do not need to find f(x) and g(x) explicitly. Hint: Consider characteristic polynomials