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

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ISBN:9781285463247

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Question

Linear algebra: please solve q19 correctly and handwritten

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
12.
-2-2
10. -7 4-3
83 3
32-16 13
6441
461
4164
14 46
4
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
23. A in Exercise 13
22. A in Exercise 9
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) = 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 × 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 (t) 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 ... a₁-ao
0
0
1
0
0
0
1
0
0
0
1
0
a) For n = 2 and for n = 3, show that
det(AtI) = (-1)" q(t).

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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