306 3. ³ [34] 2 5. [8] 3 7. [ 9. 11. -6-1 2 320 -14 -2 5 3-1-1 -12 05 -2 -1 24 24 4 01 01 3 4. 6. =] 8. 13-16 9-11 22 [33] -2 -1 0 011 -2-2-1 10. -7 4-3 83 3 32-16 13 12.6 4 4 1 461 4164 1446 Chapter 4 The Eigenvalue Problem Exercises 20-23 illustrate the Cayley-Hamilton theo- rem, which states that if p(t) is the characteristic poly- nomial for A, then p(A) is the zero matrix. (As in Ex- ercise 18, p(A) is the (n x n) matrix that comes from substituting A fort in p(t).) In Exercises 20-23, verify that p(A) = O for the given matrix A. 20. A in Exercise 3 21. A in Exercise 4 22. A in Exercise 9 23. A in Exercise 13. 16. Prove property (c) of Theorem 11. 17. Complete the proof of property (a) of Theorem 11. 18. Let q(t) = t³ -21²-t + 2; and for any (nx n) matrix H, define the matrix polynomial q (H) by q(H) = H³ - 2H² - H + 21, where I is the (n x n ) identity matrix. a) Prove that if > is an eigenvalue of H, then the number q (2) is an eigenvalue of the matrix q(H). [Hint: Suppose that Hx = λx, where x + 0, and use Theorem 11 to evaluate q(H)x.] b) Use part a) to calculate the eigenvalues of q (A) and q (B), where A and B are from Exercises 7 and 8, respectively. 19. With q (1) as in Exercise 18, verify that q (C) is the zero matrix, where C is from Exercise 9. (Note that q(t) is the characteristic polynomial for C. See Ex- ercises 20-23.) A = -an-1-an-2 1 0 ⠀ 00 0 1 .-a₁-ao 0 0 1 0 0 ⠀ 0 a) For n = 2 and for n = 3, show that