Explanation Given: The given system is X ′ = ( 0 0 0 3 0 0 5 1 0 ) 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 matrix A . A 2 = A ⋅ A = A ( 0 0 0 3 0 0 5 1 0 ) ⋅ ( 0 0 0 3 0 0 5 1 0 ) = ( 0 0 0 0 0 0 3 0 0 ) Find the matrix A 3 by multiplying matrix A 2 with matrix A . A 3 = A 2 ⋅ A = ( 0 0 0 0 0 0 3 0 0 ) ⋅ ( 0 0 0 3 0 0 5 1 0 ) = ( 0 0 0 0 0 0 0 0 0 ) Find the matrix A 4 using A 3 . A 4 = A 3 ⋅ A = ( 0 0 0 0 0 0 0 0 0 ) ⋅ ( 0 0 0 3 0 0 5 1 0 ) = ( 0 0 0 0 0 0 0 0 0 ) Proceed in this way, the general definition of a matrix raised to the power of a natural number which is given as, A k = 0 (zero matrix) for k = 3 , 4 , 5 , …

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Chapter 10.5, Problem 8E

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