For an invertible matrix A, prove that A and A-1 have the same eigenvectors. How are the eigenvalues of A related to the eigenvalues of A-1? Letting x be an eigenvector of A gives Ax = Ax for a corresponding eigenvalue 1. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = Ax Ax = Ax Ax = Ax Ax = Ax AxA-1 = AxA-1 = AA-'x XI = AA-'x x = AA-1x Ax/A = ix/A A/(Ax) = A/(1x) O(A/A)x = (A/2)x Ix = (A/1)x x = AA-1x A-'Ax = A-Ax (A/A)x = ixA-1 Ix = AxA-1 x = ixA-1 OXAA-1 Ix = 1A-1x x = JA-1x A-1x = 1x A-x = 1x A-1x = 1x A-1x = 1x This shows that ---Select-- v is an eigenvector of A- with eigenvalue ---Select- V --Select--- Need Help? 1/x 1/2

For an invertible matrix A, prove that A and A-1 have the same eigenvectors. How are the eigenvalues of A related to the eigenvalues of A-1? Letting x be an eigenvector of A gives Ax = Ax for a corresponding eigenvalue 1. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = Ax Ax = Ax Ax = Ax Ax = Ax AxA-1 = AxA-1 = AA-'x XI = AA-'x x = AA-1x Ax/A = ix/A A/(Ax) = A/(1x) O(A/A)x = (A/2)x Ix = (A/1)x x = AA-1x A-'Ax = A-Ax (A/A)x = ixA-1 Ix = AxA-1 x = ixA-1 OXAA-1 Ix = 1A-1x x = JA-1x A-1x = 1x A-x = 1x A-1x = 1x A-1x = 1x This shows that ---Select-- v is an eigenvector of A- with eigenvalue ---Select- V --Select--- Need Help? 1/x 1/2

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

For an invertible matrix A, prove that A and A- have the same eigenvectors. How are the eigenvalues of A related to the eigenvalues of A-1?
Letting x be an eigenvector of A gives Ax = Àx for a corresponding eigenvalue 1. Using matrix operations and the properties of inverse matrices gives which of the following?
Ах
Ах
= 1x
Ax = 1x
Ax = Ax
AxA“
AxA-1
Ax/A = 1x/A
A/(1x)
(A/1)x
(A/1)x
AA-1x
A/(Ax)
A-'Ax
A-12x
(A/A)x
OXAA-1
XI =
1xA-1
O(A/A)x
Ix =
Ix =
A-1x
Ix =
1xA-1
X =
X =
X =
x = 1XA-1
1x
A-1x
1x
A-1x
1x
A-1x
A-1x
1x
This shows that ---Select--- v is an eigenvector of A-- with eigenvalue ---Select---
---Select---
Need Help? 1/x
1/2

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For an invertible matrix A, prove that A and A have the same eigenvectors. How are the eigenvalues of A related to the eigenvalues of A? Letting x be an eigenvector of A gives Ax = 1x for a corresponding eigenvalue 1. Using matrix operations and the properties of inverse matrices gives which of the following? %3D Ax = Ax Ax = ix %3D Ax = 1x %3D A/(Ax) = A/(1x) Ax/A = x/A Ax = Ax AXA-1 = 1xA-1 = 1A-1x XI = A-1x x = A-1x %3D %3D %3D O(A/A)x = (A/2)x Ix = (A/2)x x = AA-1x A- Ax = A-12x Ix = A-1x x = JA-1x O(A/A)x = ixA-1 %3D %3D %3D %3D OXAA 1 Ix = %3D %3D X = XA-1 %3D %3D A-x = 1x A-1x = 1x X. %3D A-x = 1x %3D %3D A-1x = 1x %3D This shows that -Select-- vis an eigenvector of A with eigenvalue-Select--- v -Select-- Need Help? 1/x 1/2 Submit Answer
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