Explanation Given: The given system is, d x d t = 3 x − y − z d y d t = x + y − z d z d t = x − y + z Calculation: For the given system, the coefficient matrix is A = ( 3 − 1 − 1 1 1 − 1 1 − 1 1 ) . The characteristic equation for the matrix A is | ( A − λ I ) | = 0 where, I is the identity matrix. Substitute the values of matrix A and I in above expression. | 3 − λ − 1 − 1 1 1 − λ − 1 1 − 1 1 − λ | = 0 ( 3 − λ ) [ ( 1 − λ ) 2 − 1 ] + 1 [ 1 − λ + 1 ] − 1 [ − 1 + λ − 1 ] = 0 ( 3 − λ ) [ λ 2 − 2 λ ] + ( 2 − λ ) + ( 2 − λ ) = 0 ( 3 − λ ) ( λ − 2 ) λ + ( 2 − λ ) + ( 2 − λ ) = 0 Simplify further, ( λ − 2 ) [ ( 3 − λ ) λ − 1 − 1 ] = 0 ( λ − 2 ) ( 3 λ − λ 2 − 2 ) = 0 ( λ − 2 ) ( λ − 2 ) ( 1 − λ ) = 0 ( 1 − λ ) ( λ − 2 ) 2 = 0 Therefore, the eigenvalues of the above system is λ 1 = 1 with multiplicity 1 and λ 2 = 2 with multiplicity 2 . Obtain the eigenvector K 1 corresponding to the eigenvalue, λ 1 = 1 . Solve the equation ( A − λ 1 I ) K 1 = 0 where, K 1 = ( x 1 y 1 z 1 ) . Substitute the values of A , λ 1 and K 1 in the above equation gives, [ ( 3 − 1 − 1 1 1 − 1 1 − 1 1 ) − ( 1 0 0 0 1 0 0 0 1 ) ] ⋅ ( x 1 y 1 z 1 ) = ( 0 0 0 ) ( 2 − 1 − 1 1 0 − 1 1 − 1 0 ) ⋅ ( x 1 y 1 z 1 ) = ( 0 0 0 ) By applying the row transformation R 3 → R 3 − R 2 in the above system of equations, ( 2 − 1 − 1 1 0 − 1 0 − 1 1 ) ⋅ ( x 1 y 1 z 1 ) = ( 0 0 0 ) Now, by row transformation R 1 → R 1 − 2 R 2 . ( 0 − 1 1 1 0 − 1 0 − 1 1 ) ⋅ ( x 1 y 1 z 1 ) = ( 0 0 0 ) Which on applying R 1 → R 1 − R 3 in above system, ( 0 0 0 1 0 − 1 0 − 1 1 ) ⋅ ( x 1 y 1 z 1 ) = ( 0 0 0 ) ( 0 x 1 − z 1 − y 1 + z 1 ) = ( 0 0 0 ) Solve the above system of equation gives, x 1 = z 1 and y 1 = z 1 Assume that z 1 = p 1 for some scalar p 1 , which implies, x 1 = p 1 , y 1 = p 1 Thus, the value of ( x 1 y 1 z 1 ) is, ( x 1 y 1 z 1 ) = ( p 1 p 1 p 1 ) = p 1 ( 1 1 1 ) Therefore, the eigenvector K 1 corresponding to λ 1 = 1 is ( 1 1 1 )

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Author: ZILL

Publisher: CENGAGE L

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Chapter 8.2, Problem 25E

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Chapter 8.2, Problem 24E

Chapter 8.2, Problem 26E

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Chapter 8 Solutions

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Ch. 8.1 - In Problems 16 write the given linear system in...Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - In Problems 16 write the given linear system in...Ch. 8.1 - In Problems 710 write the given linear system...Ch. 8.1 - Prob. 8E Ch. 8.1 - In Problems 16 write the given linear system in...Ch. 8.1 - Prob. 10E

Ch. 8.1 - Prob. 11E Ch. 8.1 - Prob. 12E Ch. 8.1 - Prob. 13E Ch. 8.1 - In Problems 1116 verify that the vector X is a...Ch. 8.1 - Prob. 15E Ch. 8.1 - Prob. 16E Ch. 8.1 - In Problems 1720 the given vectors are solutions...Ch. 8.1 - In Problems 1720 the given vectors are solutions...Ch. 8.1 - Prob. 19E Ch. 8.1 - Prob. 20E Ch. 8.1 - Prob. 21E Ch. 8.1 - Prob. 22E Ch. 8.1 - Prob. 23E Ch. 8.1 - Prob. 24E Ch. 8.1 - Prob. 25E Ch. 8.1 - Prob. 26E Ch. 8.2 - In Problems 112 find the general solution of the...Ch. 8.2 - In Problems 112 find the general solution of the...Ch. 8.2 - Prob. 3E Ch. 8.2 - Prob. 4E Ch. 8.2 - Prob. 5E Ch. 8.2 - Prob. 6E Ch. 8.2 - Prob. 7E Ch. 8.2 - Prob. 8E Ch. 8.2 - Prob. 9E Ch. 8.2 - Distinct Real Eigenvalues In Problems 112 find the...Ch. 8.2 - Prob. 11E Ch. 8.2 - Prob. 12E Ch. 8.2 - Prob. 13E Ch. 8.2 - Prob. 14E Ch. 8.2 - In Problem 27 of Exercises 4.9 you were asked to...Ch. 8.2 - Prob. 19E Ch. 8.2 - Prob. 20E Ch. 8.2 - Prob. 21E Ch. 8.2 - Prob. 22E Ch. 8.2 - Prob. 23E Ch. 8.2 - Prob. 24E Ch. 8.2 - Prob. 25E Ch. 8.2 - Prob. 26E Ch. 8.2 - Prob. 27E Ch. 8.2 - Prob. 28E Ch. 8.2 - Prob. 29E Ch. 8.2 - Prob. 30E Ch. 8.2 - Prob. 31E Ch. 8.2 - Prob. 32E Ch. 8.2 - Prob. 33E Ch. 8.2 - Prob. 34E Ch. 8.2 - Prob. 35E Ch. 8.2 - Prob. 36E Ch. 8.2 - Prob. 37E Ch. 8.2 - Prob. 38E Ch. 8.2 - Prob. 39E Ch. 8.2 - Prob. 40E Ch. 8.2 - In Problems 3546 find the general solution of the...Ch. 8.2 - Prob. 42E Ch. 8.2 - Prob. 43E Ch. 8.2 - Prob. 44E Ch. 8.2 - In Problems 3546 find the general solution of the...Ch. 8.2 - Prob. 46E Ch. 8.2 - In Problems 47 and 48 solve the given...Ch. 8.2 - Prob. 48E Ch. 8.2 - The system of mixing tanks shown in Figure 8.2.7...Ch. 8.2 - Prob. 50E Ch. 8.2 - Prob. 51E Ch. 8.2 - Prob. 53E Ch. 8.2 - Show that the 5 5 matrix...Ch. 8.2 - Prob. 55E Ch. 8.2 - Prob. 56E Ch. 8.3 - In Problems 18 use the method of undetermined...Ch. 8.3 - Prob. 2E Ch. 8.3 - Prob. 3E Ch. 8.3 - Prob. 4E Ch. 8.3 - In Problems 18 use the method of undetermined...Ch. 8.3 - Prob. 6E Ch. 8.3 - Prob. 7E Ch. 8.3 - Prob. 8E Ch. 8.3 - Prob. 9E Ch. 8.3 - Prob. 10E Ch. 8.3 - Consider the large mixing tanks shown in Figure...Ch. 8.3 - Prob. 12E Ch. 8.3 - Prob. 13E Ch. 8.3 - Prob. 14E Ch. 8.3 - Prob. 15E Ch. 8.3 - Prob. 16E Ch. 8.3 - Prob. 17E Ch. 8.3 - Prob. 18E Ch. 8.3 - Prob. 19E Ch. 8.3 - Prob. 20E Ch. 8.3 - Prob. 21E Ch. 8.3 - Prob. 22E Ch. 8.3 - Prob. 23E Ch. 8.3 - Prob. 24E Ch. 8.3 - Prob. 25E Ch. 8.3 - Prob. 26E Ch. 8.3 - Prob. 27E Ch. 8.3 - Prob. 28E Ch. 8.3 - Prob. 29E Ch. 8.3 - Prob. 30E Ch. 8.3 - Prob. 31E Ch. 8.3 - Prob. 32E Ch. 8.3 - Prob. 33E Ch. 8.3 - Prob. 34E Ch. 8.3 - The system of differential equations for the...Ch. 8.3 - Prob. 36E Ch. 8.4 - Prob. 1E Ch. 8.4 - Prob. 2E Ch. 8.4 - Prob. 3E Ch. 8.4 - Prob. 4E Ch. 8.4 - In problem 58 use (1) use to find the general...Ch. 8.4 - In problem 58 use (1) use to find the general...Ch. 8.4 - Prob. 7E Ch. 8.4 - Prob. 8E Ch. 8.4 - Prob. 9E Ch. 8.4 - Prob. 10E Ch. 8.4 - Prob. 11E Ch. 8.4 - Prob. 12E Ch. 8.4 - Prob. 13E Ch. 8.4 - Prob. 14E Ch. 8.4 - Prob. 15E Ch. 8.4 - Prob. 16E Ch. 8.4 - In problem 1518 use the method of Example 2 to...Ch. 8.4 - Prob. 18E Ch. 8.4 - Prob. 19E Ch. 8.4 - Prob. 20E Ch. 8.4 - Prob. 21E Ch. 8.4 - Prob. 22E Ch. 8.4 - Prob. 23E Ch. 8.4 - Prob. 24E Ch. 8.4 - Prob. 25E Ch. 8.4 - Prob. 26E Ch. 8 - fill in the blanks. 1. The vector X=k(45) is a...Ch. 8 - fill in the blanks. The vector...Ch. 8 - Consider the linear system X=(466132143)X. Without...Ch. 8 - Consider the linear system X = AX of two...Ch. 8 - In Problems 514 solve the given linear system. 5....Ch. 8 - Prob. 6RE Ch. 8 - Prob. 7RE Ch. 8 - Prob. 8RE Ch. 8 - Prob. 9RE Ch. 8 - Prob. 10RE Ch. 8 - Prob. 11RE Ch. 8 - Prob. 12RE Ch. 8 - Prob. 13RE Ch. 8 - Prob. 14RE Ch. 8 - Prob. 15RE Ch. 8 - Prob. 16RE

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The general solution of the given system.

Solution Summary: The author explains that the general solution of the given system is X(t)=c_1 left, where I is the identity matrix.

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Chapter 8 Solutions