(i) Let f(a) be a polynomial such that f(A) = Onxn. Convince me that m(a) f(a). (ii) Up to similarity, classify all 5 x 5 matrices such that A - 5A = -615. [Hint: Let f(a) = a - 5a + 6. Hence, by hypothesis, f(A) = 05x5. By (i), mA(a) | f(a). Hence m(a) = a - 2 OR mA(a) = a-3 OR m(a) = f(a). ] To 0 0 1 (iii) Let A be a 4 x 4 such that A is similar to H = 00 1 0 0 0 0 1 9 - 18 8 2 a. Find CA (a) and mA (a). b. For each eigenvalue a of A find dim (Ea (A))[ Hint: C(a) = (a - 2a + 1)(a-9)] c. Theoretically, how do you construct the columns of the invertible matrix Q where Q-AQ = H?