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22. a. If A is 3 x 3, with columns a1, a2, and a3, then det A equals the volume of the parallelepiped determined by a₁, a2 and a3. b. det AT = (-1) det A. equation c. The multiplicity of a root r of the characteristic of A is called the algebraic multiplicity of r as an eigen- value of A. d. A row replacement operation on A does not change the eigenvalues.

22. a. If A is 3 x 3, with columns a1, a2, and a3, then det A equals the volume of the parallelepiped determined by a₁, a2 and a3. b. det AT = (-1) det A. equation c. The multiplicity of a root r of the characteristic of A is called the algebraic multiplicity of r as an eigen- value of A. d. A row replacement operation on A does not change the eigenvalues.

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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15. 17. 0 3 -5 0 0 0 0 -4 0 1 3 0 -7 1 A = 50x10 3-8 0 0 8 0 OOON9 18 0 0 0 1 9-2 0 0 0 0 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 0 3 0 0 0 0 3 6 -1 h 0 5 4 0 1 -5 2 - bank of tele bust 19. Let A be an n x n matrix, and suppose A has n real eigenval- ues, A₁,..., An, repeated according to multiplicities, so t det(A-AI) = (₁-2)(λ₂ λ) (and) 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. 24. 25. In Exercises 21 and 22, A and B are n x n matrices. Mark each statement True or False. Justify each answer. b. An elementary row operation on Ad determinant in A. 21. a. The determinant of A is the product of the diagonal entries 2
Transcribed Image Text:15. 17. 0 3 -5 0 0 0 0 -4 0 1 3 0 -7 1 A = 50x10 3-8 0 0 8 0 OOON9 18 0 0 0 1 9-2 0 0 0 0 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 0 3 0 0 0 0 3 6 -1 h 0 5 4 0 1 -5 2 - bank of tele bust 19. Let A be an n x n matrix, and suppose A has n real eigenval- ues, A₁,..., An, repeated according to multiplicities, so t det(A-AI) = (₁-2)(λ₂ λ) (and) 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. 24. 25. In Exercises 21 and 22, A and B are n x n matrices. Mark each statement True or False. Justify each answer. b. An elementary row operation on Ad determinant in A. 21. a. The determinant of A is the product of the diagonal entries 2
15. 17. 0 3 -5 0 0 0 0 -4 0 1 3 0 -7 1 A = 50x10 3-8 0 0 8 0 OOON9 18 0 0 0 1 9-2 0 0 0 0 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 0 3 0 0 0 0 3 6 -1 h 0 5 4 0 1 -5 2 - bank of tele bust 19. Let A be an n x n matrix, and suppose A has n real eigenval- ues, A₁,..., An, repeated according to multiplicities, so t det(A-AI) = (₁-2)(λ₂ λ) (and) 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. 24. 25. In Exercises 21 and 22, A and B are n x n matrices. Mark each statement True or False. Justify each answer. b. An elementary row operation on Ad determinant in A. 21. a. The determinant of A is the product of the diagonal entries 2
Transcribed Image Text:15. 17. 0 3 -5 0 0 0 0 -4 0 1 3 0 -7 1 A = 50x10 3-8 0 0 8 0 OOON9 18 0 0 0 1 9-2 0 0 0 0 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 0 3 0 0 0 0 3 6 -1 h 0 5 4 0 1 -5 2 - bank of tele bust 19. Let A be an n x n matrix, and suppose A has n real eigenval- ues, A₁,..., An, repeated according to multiplicities, so t det(A-AI) = (₁-2)(λ₂ λ) (and) 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. 24. 25. In Exercises 21 and 22, A and B are n x n matrices. Mark each statement True or False. Justify each answer. b. An elementary row operation on Ad determinant in A. 21. a. The determinant of A is the product of the diagonal entries 2
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