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 = ix 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 = 1x A-lAx = A-12x AXA-1 = ixA-1 Ax/A = Ax/A A/(Ax) = A/(x) OXAA-1 = 2A-lx XI = A-1x x = AA-1x O(A/A)x = ixA-1 Ix = ixA-1 Ix = AA-1x O (A/A)x = (A/2)x Ix = (A/2)x x = AA-1x A-lx = 1x x = AA-1x x = ixA-1 A-1x = 1x A-1x = 1x A-1x = 1x This shows that-Select---v is an eigenvector of A-1 with eigenvalue Select- v Select- Need Help? Read It 1/x 1/2 Submit Answer
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 = ix 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 = 1x A-lAx = A-12x AXA-1 = ixA-1 Ax/A = Ax/A A/(Ax) = A/(x) OXAA-1 = 2A-lx XI = A-1x x = AA-1x O(A/A)x = ixA-1 Ix = ixA-1 Ix = AA-1x O (A/A)x = (A/2)x Ix = (A/2)x x = AA-1x A-lx = 1x x = AA-1x x = ixA-1 A-1x = 1x A-1x = 1x A-1x = 1x This shows that-Select---v is an eigenvector of A-1 with eigenvalue Select- v Select- Need Help? Read It 1/x 1/2 Submit Answer
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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![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¯4?
Letting x be an eigenvector of A gives Ax = Ax for a corresponding eigenvalue A. Using matrix operations and the properties of inverse matrices gives which of the following?
Ax = ix
Ax = Ax
Ax = Ax
Ax
A-Ax = A-12x
AxA-1 = AxA-1
Ax/A = Ax/A
= AA-1x
XI = A-1x
x = AA-1x
O(A/A)x = AxA-1
Ix = 1xA-1
x = ixA-1
A/(Ax) = A/(ax)
O (A/A)x = (A/2)x
Ix = (A/1)x
x = 1A-1x
Ix = JA-1x
OXAA-1
x = AA-1x
A-1x = 1x
A-1x = 1x
A-1x = 1x
A-1x = 1x
This shows that
Select-- v is an eigenvector of A-1 with eigenvalue
-Select- v
Select
Need Help?
Read It
1/x
1/2
Submit Answer](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F080b5c0e-ffcb-4ace-82d3-1f50cc9b6c52%2F9828b332-5d38-4781-b02a-bfa191eb61ca%2F0l89ny_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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¯4?
Letting x be an eigenvector of A gives Ax = Ax for a corresponding eigenvalue A. Using matrix operations and the properties of inverse matrices gives which of the following?
Ax = ix
Ax = Ax
Ax = Ax
Ax
A-Ax = A-12x
AxA-1 = AxA-1
Ax/A = Ax/A
= AA-1x
XI = A-1x
x = AA-1x
O(A/A)x = AxA-1
Ix = 1xA-1
x = ixA-1
A/(Ax) = A/(ax)
O (A/A)x = (A/2)x
Ix = (A/1)x
x = 1A-1x
Ix = JA-1x
OXAA-1
x = AA-1x
A-1x = 1x
A-1x = 1x
A-1x = 1x
A-1x = 1x
This shows that
Select-- v is an eigenvector of A-1 with eigenvalue
-Select- v
Select
Need Help?
Read It
1/x
1/2
Submit Answer
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