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determinant of A is the product of the diagonal entries 21. a. The in A. b. An elementary row operation on A does not change the determinant. c. (det A) (det B) = det AB d. If λ +5 is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A.

determinant of A is the product of the diagonal entries 21. a. The in A. b. An elementary row operation on A does not change the determinant. c. (det A) (det B) = det AB d. If λ +5 is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.1: Introduction To Eigenvalues And Eigenvectors
Problem 35EQ
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17. -5 1 38 0 -7 1 -4 DOON9 A = 0 0 1 9-2 0 0 2 18. It can be shown that the algebraic multiplicity of an eigen- value is always greater than or equal to the dimension of the eigenspace corresponding to λ. Find h in the matrix A below such that the eigenspace for λ = 5 is two-dimensional: 5 -2 6 -17 0 3 h 0 4 0 0 5 0 001 0 0 0 3 19. Let A be an n x n matrix, and suppose A has n real eigenval- ues, A₁,..., An, repeated according to multiplicities, so that det(A-1) = (₁ - λ) (λ₂ λ)... (λn - λ) - Explain why det A is the product of the n eigenvalues of A. (This result is true for any square matrix when complex eigenvalues are considered.) 20. Use a property of determinants to show that A and AT have the same characteristic polynomial. In Exercises 21 and 22, A and B are n x n matrices. Mark each statement True or False. Justify each answer. 25. Le th 21. a. The determinant of A is the product of the diagonal entries in A. determinant. b. An elementary row operation on A does not change the c. (det A) (det B) = det AB then 5 is an eigenvalue of A. d. If λ + 5 is a factor of the characteristic polynomial of A, 26. a t C 27.
Transcribed Image Text:17. -5 1 38 0 -7 1 -4 DOON9 A = 0 0 1 9-2 0 0 2 18. It can be shown that the algebraic multiplicity of an eigen- value is always greater than or equal to the dimension of the eigenspace corresponding to λ. Find h in the matrix A below such that the eigenspace for λ = 5 is two-dimensional: 5 -2 6 -17 0 3 h 0 4 0 0 5 0 001 0 0 0 3 19. Let A be an n x n matrix, and suppose A has n real eigenval- ues, A₁,..., An, repeated according to multiplicities, so that det(A-1) = (₁ - λ) (λ₂ λ)... (λn - λ) - Explain why det A is the product of the n eigenvalues of A. (This result is true for any square matrix when complex eigenvalues are considered.) 20. Use a property of determinants to show that A and AT have the same characteristic polynomial. In Exercises 21 and 22, A and B are n x n matrices. Mark each statement True or False. Justify each answer. 25. Le th 21. a. The determinant of A is the product of the diagonal entries in A. determinant. b. An elementary row operation on A does not change the c. (det A) (det B) = det AB then 5 is an eigenvalue of A. d. If λ + 5 is a factor of the characteristic polynomial of A, 26. a t C 27.
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