Explanation Given: The given matrix is, T = [ 0.4 0.3 0.3 0.2 0.7 0.1 0.3 0.3 0.4 ] Approach: The equilibrium matrix L can be solved by following steps. 1. Create the transition matrix T . 2. Create the equilibrium matrix L . 3. Find and simplify the system of equations described by L T = L . 4. Discard any redundant equations, and include the equation x + y + z = 1 (or similar if there are more than three variables). 5. Solve the resulting system, using either the elimination method or the graphing calculator method. 6. Check your work by verifying that L T = L . Calculation: Consider the equilibrium matrix, L = [ x y z ] Find and simplify the system of equations described by L T = L as follows, [ x y z ] [ 0.4 0.3 0.3 0.2 0.7 0.1 0.3 0.3 0.4 ] = [ x y z ] Multiply the Matrix as follows, [ 0.4 x + 0.2 y + 0.3 z 0.3 x + 0.7 y + 0.3 z 0.3 x + 0.1 y + 0.4 z ] = [ x y z ] Equate both the side equations as follows, 0.4 x + 0.2 y + 0.3 z = x 0.3 x + 0.7 y + 0.3 z = y 0.3 x + 0.1 y + 0.4 z = z The above equations can be written as follows, − 0.6 x + 0.2 y + 0.3 z = 0 0.3 x − 0.3 y + 0.3 z = 0 0.3 x + 0.1 y − 0.6 z = 0 Multiply with 10 on the above equations. − 6 x + 2 y + 3 z = 0 …… ( 1 ) 3 x − 3 y + 3 z = 0 ..... To determine (b) To verify: The solution L T = L .

8th Edition

ISBN: 9781305104174

Author: David B. Johnson, Thomas A. Mowry

Publisher: Cengage Learning

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Chapter 11.5, Problem 2E

To determine

(a)

To find:

The equilibrium matrix L by solving LT=L for L.

To determine

(b)

To verify:

The solution LT=L.

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Chapter 11.5, Problem 1E

Chapter 11.5, Problem 3E

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(a) To find: The equilibrium matrix L by solving L T = L for L .

Solution Summary: The author explains how the equilibrium matrix L can be solved using the elimination method or the graphing calculator method.

Videos

(a)

To find:

The equilibrium matrix L by solving LT=L for L.

(b)

To verify:

The solution LT=L.

Want to see the full answer?

Chapter 11 Solutions

(a) To find: The equilibrium matrix L by solving L T = L for L .

Solution Summary: The author explains how the equilibrium matrix L can be solved using the elimination method or the graphing calculator method.

Videos

(a) To find: The equilibrium matrix L by solving LT=L for L.

(b) To verify: The solution LT=L.

Want to see the full answer?

Chapter 11 Solutions

(a)

To find:

The equilibrium matrix L by solving LT=L for L.

(b)

To verify:

The solution LT=L.