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 = ix for a corresponding eigenvalue 1. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = ix Ax = ix Ax = ix AxA-1 = ixA-1 A/ (Ax) = A / (àx) Ax /A = ix /A Ax = ix 4-1 = AA-1x A-'Ax = A-1¼x O(A/ A)x = (A / 2)x Ix = (A/ A)x O(A/ A)x = ixA-1 Ix = ixA-1 OXAA- XI = AA-1x Ix = AA-1x x = A-1x x = AA-lx 1. x = 1A-1x x = iXA-1 1x A-1x = 1x A-1x = A-1x = A-1x =

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 = ix for a corresponding eigenvalue 1. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = ix Ax = ix Ax = ix AxA-1 = ixA-1 A/ (Ax) = A / (àx) Ax /A = ix /A Ax = ix 4-1 = AA-1x A-'Ax = A-1¼x O(A/ A)x = (A / 2)x Ix = (A/ A)x O(A/ A)x = ixA-1 Ix = ixA-1 OXAA- XI = AA-1x Ix = AA-1x x = A-1x x = AA-lx 1. x = 1A-1x x = iXA-1 1x A-1x = 1x A-1x = A-1x = A-1x =

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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