Explanation Given: The given matrix is: A = [ 1 0 1 2 1 1 2 1 0 0 2 0 0 0 1 1 ] Procedure used: Compute the characteristics polynomial p ( r ) = | A − r I | . Find the zeroes of p ( r ) and express it as p ( r ) = ( r 1 − r ) m 1 ( r 2 − r ) m 2 ⋯ ( r k − r ) m k here r 1 , r 2 , ⋯ r k are the eigenvalues and m 1 , m 2 ⋯ , m k are their multiplicities. For each eigenvalue r i find m i linearly independent generalized eigenvectors by applying the Gauss-Jordan algorithm to the system ( A − r i I ) m i u = 0 . Then, assemble the fundamental matrix X ( t ) and find the general solution: x ( t ) = X 1 ( t ) + X 2 ( t ) + X 3 ( t ) + ⋯ Calculation: Consider the given matrix: A = [ 1 0 1 2 1 1 2 1 0 0 2 0 0 0 1 1 ] The characteristics polynomial is: | [ 1 0 1 2 1 1 2 1 0 0 2 0 0 0 1 1 ] − r [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] | = 0 | [ 1 0 1 2 1 1 2 1 0 0 2 0 0 0 1 1 ] − [ r 0 0 0 0 r 0 0 0 0 r 0 0 0 0 r ] | = 0 | [ 1 − r 0 1 2 1 1 − r 2 1 0 0 2 − r 0 0 0 1 1 − r ] | = 0 ( 1 − r ) 3 ( 2 − r ) = 0 Further simplify as: ( 1 − r ) 3 ( 2 − r ) = 0 r = 1 , 1 , 1 , 2 The eigenvalues are 2 and 1 with multiplicity 3. The eigenvector corresponding to the eigenvalue r = 1 : ( [ 1 0 1 2 1 1 2 1 0 0 2 0 0 0 1 1 ] − 1 [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] ) [ u 1 u 2 u 3 u 3 ] = [ 0 0 0 0 ] ( [ 0 0 1 2 1 0 2 1 0 0 1 0 0 0 1 0 ] ) [ u 1 u 2 u 3 u 3 ] = [ 0 0 0 0 ] It indicates as, u 1 = u 3 = u 4 = 0 and u 2 is free. Choose arbitrary value u 2 = 1 . The eigenvector is: v 1 = [ 0 1 0 0 ] The first solution is: X 1 = e t [ 0 1 0 0 ] We solve ( A − I ) 2 u = 0 , that is: ( [ 0 0 1 2 1 0 2 1 0 0 1 0 0 0 1 0 ] ) 2 [ u 1 u 2 u 3 u 3 ] = [ 0 0 0 0 ] ( [ 0 0 3 0 0 0 4 2 0 0 1 0 0 0 1 0 ] ) [ u 1 u 2 u 3 u 3 ] = [ 0 0 0 0 ] It indicates as, u 3 = u 4 = 0 and u 1 , u 2 are free

Pearson eText Fundamentals of Differential Equations with Boundary Value Problems -- Instant Access (Pearson+)

7th Edition

ISBN: 9780137394524

Author: R. Nagle, Edward Saff

Publisher: PEARSON+

See similar textbooks

Videos

Question

Chapter 9.8, Problem 19E

To determine

To find:

The general solution for the given matrix A=[1012112100200011].

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.8, Problem 18E

Chapter 9.8, Problem 20E

Students have asked these similar questions

Use Euler's method to numerically integrate dy dx -2x+12x² - 20x +8.5 from x=0 to x=4 with a step size of 0.5. The initial condition at x=0 is y=1. Recall that the exact solution is given by y = -0.5x+4x³- 10x² + 8.5x+1

Find an equation of the line tangent to the graph of f(x) = (5x-9)(x+4) at (2,6).

Find the point on the graph of the given function at which the slope of the tangent line is the given slope. 2 f(x)=8x²+4x-7; slope of the tangent line = -3

Chapter 9 Solutions

Pearson eText Fundamentals of Differential Equations with Boundary Value Problems -- Instant Access (Pearson+)

Knowledge Booster

Learn more about

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

The general solution for the given matrix A = [ 1 0 1 2 1 1 2 1 0 0 2 0 0 0 1 1 ] .

Videos

To find: The general solution for the given matrix A=[1012112100200011].

Want to see the full answer?

Chapter 9 Solutions

To find:

The general solution for the given matrix A=[1012112100200011].