Program Explanation Given information: The system is x ′ = [ − 2 17 4 − 1 6 1 0 1 2 ] x . The Eigen values is given as, λ = 2 , 2 , 2 Explanation: Consider matrix form of the system of equation as A . A = [ − 2 17 4 − 1 6 1 0 1 2 ] The associated Eigen vector with the matrix A is v = [ a b c ] T . Substitute 2 for λ in coefficient matrix ( A − λ I ) v = 0 , [ − 2 − 2 17 4 − 1 6 − 2 1 0 1 2 − 2 ] [ a 1 b 1 c 1 ] = [ 0 0 0 ] [ − 4 17 4 − 1 4 1 0 1 0 ] [ a 1 b 1 c 1 ] = [ 0 0 0 ] The scalar equation are shown below, − 4 a 1 + 17 b 1 + 4 c 1 = 0 − a 1 + 4 b 1 + c 1 = 0 b 1 = 0 Substitute 0 for b 1 in equation − a 1 + 4 b 1 + c 1 = 0 as, − a 1 + 4 ( 0 ) + c 1 = 0 a 1 = c 1 Consider k as a 1 then the value of b 1 and c 1 can be obtained as, b 1 = 0 and c 1 = k Now the vector can be obtained as, [ a 1 b 1 c 1 ] = [ k 0 k ] = [ − 1 0 − 1 ] k Therefore, the associated Eigen vector is obtained as: v 1 = [ − 1 0 − 1 ] The value of ( A − λ I ) 2 can be obtained as, ( A − λ I ) 2 = [ − 4 17 4 − 1 4 1 0 1 0 ] [ − 4 17 4 − 1 4 1 0 1 0 ] = [ − 1 4 1 0 0 0 − 1 4 1 ] The value of ( A − λ I ) 3 can be obtained as, ( A − λ I ) 3 = [ − 1 4 1 0 0 0 − 1 4 1 ] [ − 4 17 4 − 1 4 1 0 1 0 ] = [ 0 0 0 0 0 0 0 0 0 ] Therefore, any vector v ≠ 0 is a solution of the equation ( A − 2 I ) 3 v = 0

5th Edition

ISBN: 9780135962176

Author: Edwards

Publisher: PEARSON CO

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Question

Chapter 5.5, Problem 25P

Program Plan Intro

Program Description:Purpose of problem is to obtain the general solution of the system x′=[−2174−161012]x .

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Chapter 5.5, Problem 24P

Chapter 5.5, Problem 26P

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Chapter 5 Solutions

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Ch. 5.1 - Let A=[2347] and B=[3451]. Find (a) 2A+3B; (b)...Ch. 5.1 - Prob. 2P Ch. 5.1 - Find AB and BA given A=[203415] and B=[137032].Ch. 5.1 - Prob. 4P Ch. 5.1 - Prob. 5P Ch. 5.1 - Prob. 6P Ch. 5.1 - Prob. 7P Ch. 5.1 - Prob. 8P Ch. 5.1 - Prob. 9P Ch. 5.1 - Prob. 10P

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Program Description: Purpose of problem is to obtain the general solution of the system x ′ = [ − 2 17 4 − 1 6 1 0 1 2 ] x .

Solution Summary: The author explains that the problem is to obtain the general solution of the system of equation as A. Substitute 2 for in coefficient matrix

Program Description:Purpose of problem is to obtain the general solution of the system x′=[−2174−161012]x .

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Chapter 5 Solutions