exam2 - 3 a(6 pts Determine all values of for which the...

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3. a) (6 pts) Determine all values of for which the matrix A = 1 1 1 ± has distinct real eigenvalues. Solution: Solving det ( A rI ) = (1 r ) 2 = r 2 2 r + (1 ) = 0 we get two roots 1 ± p So we have distinct real roots if and only if & > 0 . b) (6 pts) Suppose that is among the values found in part a). Solve the linear system X 0 = AX where A = 1 1 1 ± : Solution: From part a) the eigenvalues are 1 ± p 1+ p we solve ² A ² 1 + p ³ I ³ v = 0 or p 1 + v 2 = 0 1 p 2 = 0 An eigen vector is V 1 = ( 1 p ) : 1 p we solve ² A ² 1 p ³ I ³ v = 0 or p 1 + v 2 = 0 1 + p 2 = 0 An eigen vector is V 2 = ( 1 p ) : So the general solution of the system is c 1 e (1+ p ) t V 1 + c 2 e (1 p ) t V 2 1
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1. a) (7 pts) Find the Laplace transform of the function f ( t ) = h 3 0 t < 4 ± 5 4 t < 6 e 7 t 6 t < 1 : Solution: The function f can be represented as
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This note was uploaded on 10/09/2010 for the course MATH 202 taught by Professor Şükrangüllü during the Spring '08 term at Saint Joseph's University.

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exam2 - 3 a(6 pts Determine all values of for which the...

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