Explanation Given: The given system is X ′ = ( 4 3 − 4 − 4 ) X Where the coefficient matrix is A = ( 4 3 − 4 − 4 ) . Definition used: For any matrix, the inverse of the matrix can be obtained using adjoint matrix of A defined as, A − 1 = adj A | A | Where adj A is adjoint of matrix A and | A | is the determinant of the matrix. Procedure used: In order to find the value of e A t for the given matrix A , compute the matrix ( s I − A ) and then find ( s I − A ) − 1 followed by Inverse Laplace transformation. Also, inverse Laplace transform of ( 1 s − a ) is e a t . The general solution of the system X ′ = A X is given by X = e A t C where C is constant. Calculation: The matrix is A = ( 4 3 − 4 − 4 ) Compute the value of ( s I − A ) . ( s I − A ) = s ( 1 0 0 1 ) − ( 4 3 − 4 − 4 ) = ( s − 4 − 3 4 s + 4 ) Compute the value of ( s I − A ) − 1 by using the definition of inverse of the matrix. The determinant of ( s I − A ) is computed as follows. | s I − A | = | s − 4 − 3 4 s + 4 | = s 2 − 16 + 12 = s 2 − 4 = ( s − 2 ) ( s + 2 ) And the value of adj ( s I − A ) is adj ( s I − A ) = ( s + 4 3 − 4 s − 4 ) . Therefore, the inverse ( s I − A ) − 1 can be obtained by substituting value of adj ( s I − A ) and | s I − A | in the definition of inverse as, ( s I − A ) − 1 = adj ( s I − A ) | s I − A | = 1 ( s − 2 ) ( s + 2 ) ( s + 4 3 − 4 s − 4 ) = ( s + 4 ( s − 2 ) ( s + 2 ) 3 ( s − 2 ) ( s + 2 ) − 4 ( s − 2 ) ( s + 2 ) s − 4 ( s − 2 ) ( s +<

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Chapter 10.5, Problem 15E

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To compute: The value of eAt using the coefficient matrix and then find the general solution of the given system.

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Chapter 10.5, Problem 14E

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The value of e A t using the coefficient matrix and then find the general solution of the given system.

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