Explanation Given: The given matrix is A = ( 1 0 0 2 ) . 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 + … . Calculation: Compute A 2 by multiplying the matrix A with itself. A 2 = A ⋅ A = ( 1 0 0 2 ) ⋅ ( 1 0 0 2 ) = ( 1 0 0 4 ) Obtain A 3 by multiplying the matrix A 2 with matrix . A 3 = A 2 ⋅ A = ( 1 0 0 4 ) ⋅ ( 1 0 0 2 ) = ( 1 0 0 8 ) Obtain the value of the matrix A 4 by multiply the matrix A 3 with the matrix A. A 4 = A 3 ⋅ A = ( 1 0 0 8 ) ⋅ ( 1 0 0 2 ) = ( 1 0 0 16 ) Proceed in this way to obtain the general form of a matrix raised to the power of a natural number is obtained as, A k = ( 1 0 0 2 k ) for k = 1 , 2 , 3 , … 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 obtained matrices in above expression, e A t = ( 1 0 0 1 ) + 1 1 ! ( 1 0 0 2 ) t + 1 2 ! ( 1

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ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 10.5, Problem 1E

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

Solution Summary: The author calculates A2 by multiplying the matrix with itself.

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

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