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?%3DAx = AxAx = ix%3DAx = 1x%3DA/(Ax) = A/(1x)Ax/A = x/AAx = AxAXA-1 = 1xA-1= 1A-1xXI = A-1xx = A-1x%3D%3D%3DO(A/A)x = (A/2)xIx = (A/2)xx = AA-1xA- Ax = A-12xIx = A-1xx = JA-1xO(A/A)x = ixA-1%3D%3D%3D%3DOXAA 1Ix =%3D%3DX = XA-1%3D%3DA-x = 1xA-1x = 1xX.%3DA-x = 1x%3D%3DA-1x = 1x%3DThis shows that -Select-- vis an eigenvector of A with eigenvalue-Select--- v-Select--Need Help?1/x1/2Submit Answer

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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 = Ax for a corresponding eigenvalue 2. Using matrix operations and the preperties of inverse matrices gives which of the following? Ax = Ax Ax = x Ax = ix AXA-1 = 2xA-1 = 2A-1x xI = JA-1x A/(Ax) = A/(Ax) Ax/A = Ax/A Ax = ix OA/A)x = (A/A)x Ix = (A/A)x O(A/A)x = ixA-1 Ix = 2xA-1 A-Ax = A-12x Ix = A-1x x = 1A-1x x = JA-x x= xA-1 A-x = 1x A-x = 1x A-x = 1x A-x = 1x This shows that-Select-- vis an eigenvector of A with eigenvalue -Select--v Select Need Help? 1/x 1/A 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 = Ax for a corresponding eigenvalue 2. Using matrix operations and the preperties of inverse matrices gives which of the following? Ax = Ax Ax = x Ax = ix AXA-1 = 2xA-1 = 2A-1x xI = JA-1x A/(Ax) = A/(Ax) Ax/A = Ax/A Ax = ix OA/A)x = (A/A)x Ix = (A/A)x O(A/A)x = ixA-1 Ix = 2xA-1 A-Ax = A-12x Ix = A-1x x = 1A-1x x = JA-x x= xA-1 A-x = 1x A-x = 1x A-x = 1x A-x = 1x This shows that-Select-- vis an eigenvector of A with eigenvalue -Select--v Select Need Help? 1/x 1/A Submit Answer

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.3: Eigenvalues And Eigenvectors Of N X N Matrices

Problem 41EQ

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