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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 .....
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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.
 Spring '11
 Francis

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