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 = ix Ax = Ax A/(Ax) = A/(Ax) O(A/A)x = (A/2)x Ix = (A/2)x x = AA-1x A-'Ax = A-1x Ax/A = Ax/A AxA-1 = ixA-1 O (A/A)x = ixA-1 Ix = AA-1x x = AA-1x OXAA-1 XI = AA-1x x = AA-1x Ix = ixA-1 x = AxA-1 A-lx = 1x %3D A-1x = 1x A-1x = 1x A-1x = 1x This shows that ---Select--- is an eigenvector of A-1 with eigenvalue --Select-

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 = ix Ax = Ax A/(Ax) = A/(Ax) O(A/A)x = (A/2)x Ix = (A/2)x x = AA-1x A-'Ax = A-1x Ax/A = Ax/A AxA-1 = ixA-1 O (A/A)x = ixA-1 Ix = AA-1x x = AA-1x OXAA-1 XI = AA-1x x = AA-1x Ix = ixA-1 x = AxA-1 A-lx = 1x %3D A-1x = 1x A-1x = 1x A-1x = 1x This shows that ---Select--- is an eigenvector of A-1 with eigenvalue --Select-

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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