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(a) Let N E R. Consider the matrices cos(2) - sin(2) ) sin(N) cos(2) cos(N) sin(N) sin(N) - cos(SN) A(N) B(N) Find the determinant of these matrices. Show that for the matrix A(N) the eigenvalues depend on N but the eigenvectors do not. Show that for the matrix B() the eigenvalues do not depend on N but the eigenvectors do.

(a) Let N E R. Consider the matrices cos(2) - sin(2) ) sin(N) cos(2) cos(N) sin(N) sin(N) - cos(SN) A(N) B(N) Find the determinant of these matrices. Show that for the matrix A(N) the eigenvalues depend on N but the eigenvectors do not. Show that for the matrix B() the eigenvalues do not depend on N but the eigenvectors do.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(a) Let 2 E R. Consider the matrices cos(N) – sin(2) ), B(M)= ( sin(2) - cos(2) ) sin(N) cos(N) cos(2) sin(N) A(1) = ( COS COS Find the determinant of these matrices. Show that for the matrix A(2) the eigenvalues depend on 2 but the eigenvectors do not. Show that for the matrix B() the eigenvalues do not depend on 2 but the eigenvectors do. (b) Let N E R. Consider the matrices cosh(2) – sinh(2) C(?) = ( sinh(2) cosh(2) )· D(!) = ( sinh(2) - cosh(N) ) cosh(N) sinh(N) Find the determinant of these matrices. Show that for the matrix C(2) the eigenvalues depend on 2 but the eigenvectors do not. Show that for the matrix D() the eigenvalues do not depend on 2 but the eigenvectors do. (c) Show that the matrix cos e – sin 0 cos O A = sin 0 will have complex eigenvalues if 0 is not a multiple of T. Give a geometric interpretation of this result.
Transcribed Image Text:(a) Let 2 E R. Consider the matrices cos(N) – sin(2) ), B(M)= ( sin(2) - cos(2) ) sin(N) cos(N) cos(2) sin(N) A(1) = ( COS COS Find the determinant of these matrices. Show that for the matrix A(2) the eigenvalues depend on 2 but the eigenvectors do not. Show that for the matrix B() the eigenvalues do not depend on 2 but the eigenvectors do. (b) Let N E R. Consider the matrices cosh(2) – sinh(2) C(?) = ( sinh(2) cosh(2) )· D(!) = ( sinh(2) - cosh(N) ) cosh(N) sinh(N) Find the determinant of these matrices. Show that for the matrix C(2) the eigenvalues depend on 2 but the eigenvectors do not. Show that for the matrix D() the eigenvalues do not depend on 2 but the eigenvectors do. (c) Show that the matrix cos e – sin 0 cos O A = sin 0 will have complex eigenvalues if 0 is not a multiple of T. Give a geometric interpretation of this result.
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