Explanation Given: The given matrix A is, A = [ − 16 30 − 18 − 8 8 16 8 − 15 9 ] Approach: Solve the vector differential equation x ' = A x in the case where A is a defective matrix by constructing cycles of generalized eigenvectors and forming corresponding solutions x k ( t ) as, Let, A be an n × n matrix. Suppose the eigenvalue λ has multiplicity m and that the dimension of the corresponding eigenspace is r ≤ m . Then there exist m linearly independent solutions to x ' = A x that can be constructed in r cycles (one for each i with 1 ≤ i ≤ r ) of the form. x 0 ( i ) ( t ) = e λ t v 0 ( i ) , x 1 ( i ) ( t ) = e λ t ( v 1 ( i ) + t v 0 ( i ) ) x 2 ( i ) ( t ) = e λ t ( v 2 ( i ) + t v 1 ( i ) + 1 2 ! t 2 v 0 ( i ) ) ⋮ x k i ( i ) ( t ) = e λ t ( v k i ( i ) ) + t v k i − 1 ( i ) + ⋯ + 1 k ! t k i v 0 ( i ) Where each k i ≥ 0 , and k 1 + k 2 + ⋯ + k r = m − r . Calculation: Consider the given matrix A , A = [ − 16 30 − 18 − 8 8 16 8 − 15 9 ] The characteristic equation of A is given as det ( A − λ I ) = 0 , det ( A − λ I ) = 0 | [ − 16 30 − 18 − 8 8 16 8 − 15 9 ] − [ λ 0 0 0 λ 0 0 0 λ ] | = 0 | [ − 16 − λ 30 − 18 − 8 8 − λ 16 8 − 15 9 − λ ] | = 0 Solve the above equation for the eigenvalues ( λ ) of A , | − 16 − λ 30 − 18 − 8 8 − λ 16 8 − 15 9 − λ | = 0 The eigenvalues of A are 8 , − 7 , 0 . The eigenvector corresponding to the eigenvalue λ 1 = 8 is determined based on the linear system given as, ( A − λ I ) v = 0 ( [ − 16 30 − 18 − 8 8 16 8 − 15 9 ] − [ λ 0 0 0 λ 0 0 0 λ ] ) [ v 1 v 2 v 3 ] = [ 0 0 0 ] [ − 16 − λ 30 − 18 − 8 8 − λ 16 8 − 15 9 − λ ] [ v 1 v 2 v 3 ] = [ 0 0 0 ] Substitute 8 for λ in the above equation. [ − 16 − ( 8 ) 30 − 18 − 8 8 − ( 8 ) 16 8 − 15 9 − ( 8 ) ] [ v 1 v 2 v 3 ] = [ 0 0 0 ] [ − 16 − 8 30 − 18 − 8 8 − 8 16 8 − 15 9 − 8 ] [ v 1 v 2 v 3 ] = [ 0 0 0 ] [ − 24 30 − 18 − 8 0 16 8 − 15 1 ] [ v 1 v 2 v 3 ] = [ 0 0 0 ] The augmented matrix of the above matrix is, [ − 24 30 − 18 − 8 0 16 8 − 15 1 | 0 0 0 ] Apply R 1 → 1 ( − 3 ) R 1 , [ 8 − 10 6 − 8 0 16 8 − 15 1 | 0 0 0 ] Apply R 3 → R 3 + R 2 , [ 8 − 10 6 − 8 0 16 0 − 15 17 | 0 0 0 ] Apply R 1 → R 1 + R 2 , [ 0 − 10 22 − 8 0 16 0 − 15 17 | 0 0 0 ] Apply R 2 → 2 R 2 + R 3 , [ 0 − 10 22 − 8 0 16 0 0 − 32 | 0 0 0 ] Apply R 2 → 2 R 2 + R 3 , [ 0 − 10 22 − 16 0 0 0 0 − 32 | 0 0 0 ] Apply R 2 → ( − 1 16 ) R 2 , R 3 → − 1 32 ( R 3 ) and R 1 → 1 2 ( R 1 ) , [ 0 − 5 11 1 0 0 0 0 1 | 0 0 0 ] Thus, [ 0 − 5 11 1 0 0 0 0 1 ] [ v 1 v 2 v 3 ] = [ 0 0 0 ] [ − 5 v 2 + 11 v 3 v 1 v 3 ] = [ 0 0 0 ] − 5 v 2 + 11 v 3 = 0 v 1 = 0 v 3 = 0 For v 3 = 0 then v 2 = 0 . Thus, v 1 = 0 , v 2 = 0 and v 3 = 0 , the solution to this system is, v = [ v 1 , v 2 , v 3 ] T = [ 0 , 0 , 0 ] T = [ 0 0 0 ] The eigenvector for λ 1 = 8 is v 1 = [ 0 0 0 ] . Thus, x 1 ( t ) = e 8 t [ 0 0 0 ] = 0 The eigenvector corresponding to the eigenvalue λ 2 = − 7 is determined based on the linear system given as, ( A − λ I ) v = 0 ( [ − 16 30 − 18 − 8 8 16 8 − 15 9 ] − [ λ 0 0 0 λ 0 0 0 λ ] ) [ v 1 v 2 v 3 ] = [ 0 0 0 ] [ − 16 − λ 30 − 18 − 8 8 − λ 16 8 − 15 9 − λ ] [ v 1 v 2 v 3 ] = [ 0 0 0 ] Substitute − 7 for λ in the above equation,

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 9.11, Problem 16AP

To determine

The general solution to the system x'=Ax for the matrix A=[−1630−18−88168−159].

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.11, Problem 15AP

Chapter 9.11, Problem 17AP

Students have asked these similar questions

Co Given show that Solution Take home Су-15 1994 +19 09/2 4 =a log суто - 1092 ж = a-1 2+1+8 AI | SHOT ON S4 INFINIX CAMERA

a Question 7. If det d e f ghi V3 = 2. Find det -1 2 Question 8. Let A = 1 4 5 0 3 2. 1 Find adj (A) 2 Find det (A) 3 Find A-1 2g 2h 2i -e-f -d 273 2a 2b 2c

Question 1. Solve the system - x1 x2 + 3x3 + 2x4 -x1 + x22x3 + x4 2x12x2+7x3+7x4 Question 2. Consider the system = 1 =-2 = 1 3x1 - x2 + ax3 = 1 x1 + 3x2 + 2x3 x12x2+2x3 = -b = 4 1 For what values of a, b will the system be inconsistent? 2 For what values of a, b will the system have only one solution? For what values of a, b will the saystem have infinitely many solutions?

Chapter 9 Solutions

Differential Equations and Linear Algebra (4th Edition)

Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - Prob. 5TFR Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - Prob. 7TFR Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - Prob. 9TFR Ch. 9.1 - Prob. 10TFR

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

The general solution to the system x ' = A x for the matrix A = [ − 16 30 − 18 − 8 8 16 8 − 15 9 ] .

Solution Summary: The author explains how to solve the vector differential equation x'=Ax for the given matrix A.

Videos

The general solution to the system x'=Ax for the matrix A=[−1630−18−88168−159].

Want to see the full answer?

Chapter 9 Solutions