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 = ix for a corresponding eigenvalue 2. Using matrix operations and the properties of inverse matrices gives which of the following Ax = ix Ax = Ax Ax = ix Ax = Ax A-1Ax = A-12x Ax/A = Ax/A A/(Ax) = A/(ax) AxA-1 = ixA-1 O(A/A)x = ixA-1 Ix = ixA-1 O(A/A)x = (A/a)x Ix = (A/1)x x = AA-1x OXAA-1 = AA-1x Ix = AA-1x x = AA-1x 1x xI - AA-1x x = ixA-1 x = AA-1x A-1x = A-1x = 1x A-1x = 1x A-1x = 1x This shows that --Select-- v is an eigenvector of A-1 with eigenvalue --Select--- ♥ -Select- Need Help? Read It 1/x

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 = ix for a corresponding eigenvalue 2. Using matrix operations and the properties of inverse matrices gives which of the following Ax = ix Ax = Ax Ax = ix Ax = Ax A-1Ax = A-12x Ax/A = Ax/A A/(Ax) = A/(ax) AxA-1 = ixA-1 O(A/A)x = ixA-1 Ix = ixA-1 O(A/A)x = (A/a)x Ix = (A/1)x x = AA-1x OXAA-1 = AA-1x Ix = AA-1x x = AA-1x 1x xI - AA-1x x = ixA-1 x = AA-1x A-1x = A-1x = 1x A-1x = 1x A-1x = 1x This shows that --Select-- v is an eigenvector of A-1 with eigenvalue --Select--- ♥ -Select- Need Help? Read It 1/x

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 24EQ

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