(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

This is the second part of a multi-part question. Part (a) shows that:

x'' =Ax= [tex]\left(\stackrel{-2}{4/3}\stackrel{3/2}{-3}\right)[/tex]x

Part (b): Assumex= [tex]\epsilon[/tex]e[tex]^{rt}[/tex] and show that (A- r[tex]^{2}[/tex]I)[tex]\epsilon[/tex] =0

xis the solution to the second order differential equation above, and [tex]\epsilon[/tex] is an eigenvector corresponding to the eigenvalue r[tex]^{2}[/tex] ofA.

3. The attempt at a solution

Part (b):Alright, so given the above, I stated that [tex]\epsilon[/tex] = e[tex]^{-rt}[/tex]x

I then substituted everything into the left side of the equation I'm trying to prove to obtain:

(A- r[tex]^{2}[/tex]I)[tex]\epsilon[/tex] = [tex]\left(\stackrel{-2-r^{2}}{4/3}\stackrel{3/2}{-3-r^{2}}\right)[/tex][tex]\left(\stackrel{x_{1}}{x_{2}}\right)[/tex]e[tex]^{-rt}[/tex]

from here, I can see that I am not going in the right direction... any suggestions to get me moving along?

Thank-you!

-J

EDIT: for clarification, the above are 2x2 matrices... latex put the entries fairly close together. The matrix A has entries (-2 3/2) on the top and (4/3 -3) on the bottom.

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# Homework Help: Complex Eigenvalues - Proof

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