306 3. ³ [34] $[6] 5. 1 -1 3 7. 9. 11. -6-1 2 320 -14 -2 5 3-1-1 -12 0 5 4-2-1 24 4 1-1 3 4. 13-16 9-11 6. 8. 2 2 33 -2 -2 -1 10. -7 4-3 83 3 32 16 13 12. 6441 461 4 416 4 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 × 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 with Ax = λx, x = 0.] 16. Prove property (c) of Theorem 11. 17. Complete the proof of property (a) of Theorem 11. 18. Let q(t) = 1³-21²-1 + 2; and for any (n = n) matrix H, define the matrix polynomial q (H) by q(H) = H³ - 2H² H +21, where I is the (nx n) identity matrix. a) Prove that if λ 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.) -an-1-an-2 ... a₁-ao 1 0 0 0 0 1 0 0 00 0 0 1 0 A = a) For n = 2 and for n = 3, show that det(AtI) = (-1)"q(t).