Explanation Given: The given matrix is, T = [ 0.3 0.2 0.5 0.1 0.8 0.1 0.4 0.3 0.3 ] 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 ] [ 0.3 0.2 0.5 0.1 0.8 0.1 0.4 0.3 0.3 ] = [ x y z ] Multiply the Matrix as follows, [ 0.3 x + 0.1 y + 0.4 z 0.2 x + 0.8 y + 0.3 z 0.5 x + 0.1 y + 0.3 z ] = [ x y z ] Equate both the side equations as follows, 0.3 x + 0.1 y + 0.4 z = x 0.2 x + 0.8 y + 0.3 z = y 0.5 x + 0.1 y + 0.3 z = z The above equations can be written as follows, − 0.7 x + 0.1 y + 0.4 z = 0 0.2 x − 0.2 y + 0.3 z = 0 0.5 x + 0.1 y − 0.7 z = 0 Multiply with 10 to the above equations. − 7 x + 1 y + 4 z = 0 …… ( 1 ) 2 x − 2 y + 3 z = 0 ...... ( 2 ) 5 x + 1 y − 7 z = 0 …… ( 3 ) L = [ x y z ] is a probability equation so sum of its entries must be equal to 1 that means, x + y + z = 1 …… ( 4 ) Multiply with − 2 in equation ( 4 ) and add the results with equation ( 2 ) as follows, 2 x − 2 y + 3 z = 0 − 2 x − 2 y − 2 z = − 2 _ To determine (b) To verify: The solution L T = L .

MATHEMATICS A PRACTICAL ODYSSEY W/ACCESS

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ISBN: 9780357537343

Author: Johnson

Publisher: CENGAGE L

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Chapter 11.5, Problem 1E

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 11.4, Problem 28E

Chapter 11.5, Problem 2E

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Consider the cones K = = {(x1, x2, x3) | € R³ : X3 ≥√√√2x² + 3x² M = = {(21,22,23) (x1, x2, x3) Є R³: x3 > + 2 3 Prove that M = K*. Hint: Adapt the proof from the lecture notes for finding the dual of the Lorentz cone. Alternatively, prove the formula (AL)* = (AT)-¹L*, for any cone LC R³ and any 3 × 3 nonsingular matrix A with real entries, where AL = {Ax = R³ : x € L}, and apply it to the 3-dimensional Lorentz cone with an appropriately chosen matrix A.

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MATHEMATICS A PRACTICAL ODYSSEY W/ACCESS

Ch. 11.0A - In Exercises 1-10, a find the dimensions of the...Ch. 11.0A - Prob. 2E Ch. 11.0A - Prob. 3E Ch. 11.0A - Prob. 4E Ch. 11.0A - Prob. 5E Ch. 11.0A - Prob. 6E Ch. 11.0A - Prob. 7E Ch. 11.0A - Prob. 8E Ch. 11.0A - Prob. 9E Ch. 11.0A - In Exercises 1-10, a find the dimensions of the...

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

Solution Summary: The author explains how to find the equilibrium matrix L by solving LT=L for L.

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(a)

To find:

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

(b)

To verify:

The solution LT=L.

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Chapter 11 Solutions

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

Solution Summary: The author explains how to find the equilibrium matrix L by solving LT=L for L.

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.