# ps4-soln - Problem Set 4 Solutions February 3 2010 Problem...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Problem Set 4 Solutions February 3, 2010 Problem 1 Compute e At using the Laplace transform method and the eigenvalue/eigenvector method for the following matrix: A = - 2- 2 1- 3- 4 . Using the Laplace transform method, we must compute ( sI- A )- 1 . We have: ( sI- A ) = s + 2 2 s- 1 3 s + 4 . We have that det( sI- A ) = ( s + 2)( s + 1)( s + 3) and hence, ( sI- A )- 1 = 1 ( s + 2)( s + 1)( s + 3) ( s + 1)( s + 3)- 2( s + 4)- 2 ( s + 2)( s + 4) ( s + 2)- 3( s + 2) s ( s + 2) . Next, we take the inverse Laplace transform of each entry. Firstly, we must simplify some of the entries using partial fraction expansions. We have- 2( s + 4) ( s + 2)( s + 1)( s + 3) = 4 s + 2- 3 s + 1- 1 s + 3- 2 ( s + 2)( s + 1)( s + 3) = 2 s + 2- 1 s + 1- 1 s + 3 s + 4 ( s + 1)( s + 3) = 3 / 2 s + 1- 1 / 2 s + 3 1 ( s + 1)( s + 3) = 1 / 2 s + 1- 1 / 2 s + 3- 3 ( s + 1)( s + 3) =- 3 / 2 s + 1 + 3 / 2 s + 3 s ( s + 1)( s + 3) =- 1 / 2 s + 1 + 3 / 2 s + 3 . Lastly, we have e At equals the inverse Laplace transform of the entries of ( sI- A )- 1 . Hence, e At = e- 2 t 4 e- 2 t- 3 e- t- e- 3 t 2 e- 2 t- e- t- e- 3 t 3 2 e- t- 1 2 e- 3 t 1 2 e- t- 1 2 e- 3 t- 3 2 e- t + 3 2 e- 3 t- 1 2 e- t + 3 2 e- 3 t .....
View Full Document

## This note was uploaded on 01/30/2012 for the course ECE ECE311 taught by Professor Francis during the Spring '11 term at University of Toronto.

### Page1 / 5

ps4-soln - Problem Set 4 Solutions February 3 2010 Problem...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online