Explanation Given: The given matrix is A = ( 0 1 1 0 ) . Concept: For any matrix A of order n × n , the expansion of e A t in power of the matrix A is e A t = 1 + A t + A 2 2 ! t 2 + A 3 3 ! t 3 + … + A k k ! t k + … Calculation: Compute A 2 by multiplying matrix A with itself. A 2 = A ⋅ A = ( 0 1 1 0 ) ⋅ ( 0 1 1 0 ) = ( 1 0 0 1 ) = I Multiply matrix A 2 with matrix A to obtain A 3 . A 3 = A 2 ⋅ A = ( 0 1 1 0 ) = A Multiply the matrix A 3 with the matrix A to obtain the value of the matrix A 4 . A 4 = A 3 ⋅ A = ( 1 0 0 1 ) = I Proceed in the similar way, the general value of a matrix raised to the power of a natural number is obtained as, A k = { A k = 1 , 3 , 5 , … I k = 2 , 4 , 6 , … Thus, the value of the expansion e A t in power of the matrix A is e A t = 1 + A 1 ! t + A 2 2 ! t 2 + A 3 3 ! t 3 + … Substitute the value of matrices in above expression, e A t = I + A 1 ! t + I 2 ! t 2 + A 3 ! t 3 + … = I ( 1 + 1 2 ! t 2 + 1 4 ! t 4 + … ) + A ( t + 1 3 ! t 3 + 1 5 ! t 5</

Bundle: A First Course in Differential Equations with Modeling Applications, Loose-leaf Version, 11th + WebAssign Printed Access Card for Zill's ... Problems, 9th Edition, Single-Term

11th Edition

ISBN: 9781337761000

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 8.4, Problem 2E

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To find: The value of eAt and e−At using the given matrix A.

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The value of e A t and e − A t using the given matrix A .

Solution Summary: The author calculates the value of eAt and -a using the given matrix.

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To find: The value of eAt and e−At using the given matrix A.

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