15. (i) Let A, B be 2 × 2 matrices over R and vectors x, y in R2such that Ax = y, By = x, x"y = 0 and x"x = 1, y"y = 1. Show that ABand BA have an eigenvalue +1.„T,%3D(ii) Find all 2 × 2 matrices A, B which satisfy the conditions given in (i). Usex =sin(a) )– sin(a) )cos(a)y =

Answered: Image /qna-images/answer/04e38118-f5f2-42d0-96da-3c696dd567cf.jpg

15. (i) Let A, B be 2 x 2 matrices over R and vectors x, y in R2 such that Ax = y, By = x, x"y = 0 and x"x = 1, y"y = 1. Show that AB and BA have an eigenvalue +1. %3D %3D (ii) Find all 2 × 2 matrices A, B which satisfy the conditions given in (i). Use cos(a) x = - sin(a)) cos(a) y = sin(o)),

15. (i) Let A, B be 2 x 2 matrices over R and vectors x, y in R2 such that Ax = y, By = x, x"y = 0 and x"x = 1, y"y = 1. Show that AB and BA have an eigenvalue +1. %3D %3D (ii) Find all 2 × 2 matrices A, B which satisfy the conditions given in (i). Use cos(a) x = - sin(a)) cos(a) y = sin(o)),

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 70EQ

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Question

15. (i) Let A, B be 2 × 2 matrices over R and vectors x, y in R2
such that Ax = y, By = x, x"y = 0 and x"x = 1, y"y = 1. Show that AB
and BA have an eigenvalue +1.
„T,
%3D
(ii) Find all 2 × 2 matrices A, B which satisfy the conditions given in (i). Use
x =
sin(a) )
– sin(a) )
cos(a)
y =

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