Explanation Given: The given system is X ′ = ( 1 1 1 1 1 1 − 2 − 2 − 2 ) X . Definition used: 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 + … . The general solution of the system X ′ = AX is given by X = e At C where C is constant. Calculation: Compute A 2 by multiplying matrix A with itself. A 2 = A ⋅ A = ( 1 1 1 1 1 1 − 2 − 2 − 2 ) ⋅ ( 1 1 1 1 1 1 − 2 − 2 − 2 ) = ( 0 0 0 0 0 0 0 0 0 ) Multiply matrix A 2 with matrix A to obtain A 3 . A 3 = A 2 ⋅ A = ( 0 0 0 0 0 0 0 0 0 ) Multiply the matrix A 3 with the matrix A to obtain the value of the matrix A 4 . A 4 = A 3 ⋅ A = ( 0 0 0 0 0 0 0 0 0 ) Proceed in this way, the general value of a matrix raised to the power of a natural number is obtained as, A k = 0 , the zero matrix for k = 2 , 3 , 4 , …

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ISBN: 9781337652476

Author: ZILL

Publisher: CENGAGE L

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Chapter 8.4, Problem 7E

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To find: The general solution of the given system using matrix A.

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The general solution of the given system using matrix A .

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