Explanation Given: The given matrix is: A = [ 0 1 0 0 0 1 1 − 1 1 ] Formula used: Compute the characteristics polynomial p ( r ) = | A − r I | . Find the zeroes of p ( r ) and express it as p ( r ) = ( r 1 − r ) m 1 ( r 2 − r ) m 2 ⋯ ( r k − r ) m k here r 1 , r 2 , ⋯ r k are the eigenvalues and m 1 , m 2 ⋯ , m k are their multiplicities. For each eigenvalue r i find m i linearly independent generalized eigenvectors by applying the Gauss-Jordan algorithm to the system ( A − r i I ) m i u = 0 . Then, assemble the fundamental matrix X ( t ) and use the formula: e A t = X ( t ) X ( 0 ) − 1 Calculation: Consider the given matrix: A = [ 0 1 0 0 0 1 1 − 1 1 ] The characteristics polynomial is: | [ 0 1 0 0 0 1 1 − 1 1 ] − r [ 1 0 0 0 1 0 0 0 1 ] | = 0 | [ 0 1 0 0 0 1 1 − 1 1 ] − [ r 0 0 0 r 0 0 0 r ] | = 0 | [ − r 1 0 0 − r 1 1 − 1 1 − r ] | = 0 − r ⋅ [ − r 1 − 1 1 − r ] − 1 ⋅ [ 0 1 1 1 − r ] = 0 Further simplify as: − r ⋅ [ − r ( 1 − r ) + 1 ] + 1 = 0 ( r − 1 ) ( r 2 + 1 ) = 0 r = 1 , ± i The eigenvalues are 1 and ± i . The eigenvector corresponding to the eigenvalue r = 1 : ( [ 0 1 0 0 0 1 1 − 1 1 ] − 1 [ 1 0 0 0 1 0 0 0 1 ] ) [ u 1 u 2 u 3 ] = [ 0 0 0 ] ( [ 0 1 0 0 0 1 1 − 1 1 ] − [ 1 0 0 0 1 0 0 0 1 ] ) [ u 1 u 2 u 3 ] = [ 0 0 0 ] [ − 1 1 0 0 − 1 1 1 − 1 0 ] [ u 1 u 2 u 3 ] = [ 0 0 0 ] It indicates as, − 2 u 1 + u 2 = 0 and − u 2 + u 3 = 0 .Choose u 3 = 1 , then u 1 = 1 , u 2 = 1 . The eigenvector is: v 1 = [ 1 1 1 ] The first solution is: X 1 = e t [ 1 1 1 ] The eigenvector corresponding to the eigenvalue r = i : ( [ 0 1 0 0 0 1 1 − 1 1 ] − i [ 1 0 0 0 1 0 0 0 1 ] ) [ u 1 u 2 u 3 ] = [ 0 0 0 ] ( [ 0 1 0 0 0 1 1 − 1 1 ] − [ i 0 0 0 i 0 0 0 i ] ) [ u 1 u 2 u 3 ] = [ 0 0 0 ] [ − i 1 0 0 − i 1 1 − 1 1 − i ] [ u 1 u 2 u 3 ] = [ 0 0 0 ] It indicates as, − i u 1 + u 2 = 0 and − i u 2 + u 3 = 0

Fundamentals Of Differential Equations And Boundary Value Problems, Books A La Carte Edition (7th Edition)

7th Edition

ISBN: 9780321977182

Author: Nagle, R. Kent, Saff, Edward B., Snider, Arthur David

Publisher: PEARSON

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Chapter 9.8, Problem 9E

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The matrix eAt by finding the fundamental matrix X(t) for x′=Ax, for the given matrix A=[0100011−11].

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Chapter 9.8, Problem 8E

Chapter 9.8, Problem 10E

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Fundamentals Of Differential Equations And Boundary Value Problems, Books A La Carte Edition (7th Edition)

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The matrix e A t by finding the fundamental matrix X ( t ) for x ′ = A x , for the given matrix A = [ 0 1 0 0 0 1 1 − 1 1 ] .

Solution Summary: The author explains how to determine the matrix eAt by finding the fundamental matrix for the given matrix.

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The matrix eAt by finding the fundamental matrix X(t) for x′=Ax, for the given matrix A=[0100011−11].

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Chapter 9 Solutions