Explanation Given: The given matrix is: A = [ 1 1 4 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 = [ 1 1 4 1 ] The characteristics polynomial is: | [ 1 1 4 1 ] − r [ 1 0 0 1 ] | = 0 | [ 1 1 4 1 ] − [ r 0 0 r ] | = 0 | [ 1 − r 1 4 1 − r ] | = 0 ( 1 − r ) 2 − 4 = 0 Further simplify as: ( r − 3 ) ( r + 1 ) = 0 r = 3 , − 1 The eigenvalues are 3 and − 1 . The eigenvector corresponding to the eigenvalue r = 3 : ( [ 1 1 4 1 ] − 3 [ 1 0 0 1 ] ) [ u 1 u 2 ] = [ 0 0 ] ( [ 1 1 4 1 ] − [ 3 0 0 3 ] ) [ u 1 u 2 ] = [ 0 0 ] [ − 2 1 4 − 2 ] [ u 1 u 2 ] = [ 0 0 ] It indicates as, − 2 u 1 + u 2 = 0 . Choose u 2 = 2 , then u 1 = 1 . The eigenvector is: v 1 = [ 1 2 ] The first solution is: X 1 = e 3 t [ 1 2 ] The eigenvector corresponding to the eigenvalue r = − 1 :

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

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ISBN: 9780321977182

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

Publisher: PEARSON

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Chapter 9.8, Problem 8E

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The matrix eAt by finding the fundamental matrix X(t) for x′=Ax, for the given matrix A=[1141].

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Chapter 9.8, Problem 7E

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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 = [ 1 1 4 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=[1141].

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