Explanation Given: From the given data matrix is, T = [ 0.71 0.12 0.17 0.32 0.34 0.34 0.02 0.02 0.96 ] [ S F T ] 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 ] [ 0.71 0.12 0.17 0.32 0.34 0.34 0.02 0.02 0.96 ] = [ x y z ] Multiply the Matrix as follows, [ 0.71 x + 0.32 y + 0.02 z 0.12 x + 0.34 y + 0.02 z 0.17 x + 0.34 y + 0.96 z ] = [ x y z ] Equate both the side equations as follows, 0.71 x + 0.32 y + 0.02 z = x 0.12 x + 0.34 y + 0.02 z = y 0.17 x + 0.34 y + 0.96 z = z The above equations can be written as follows, − 0.29 x + 0.32 y + 0.02 z = 0 0.12 x − 0.66 y + 0.02 z = 0 0.17 x + 0.34 y − 0.04 z = 0 Multiply with 100 to the above equations. − 29 x + 32 y + 2 z = 0 …… ( 1 ) 12 x − 66 y + 2 z = 0 ...... ( 2 ) 17 x + 34 y − 4 z = 0 …… ( 3 ) L = [ x y z ] is a probability equation so sum of its entries must be equal to 1 that mean, x + y + z = 1 …… ( 4 ) Multiply with − 12 in equation ( 4 ) and add the results with equation ( 2 ) as follows, 12 x − 66 y + 2 z = 0 − 12 x − 12 y − 12 z = − 12

MATHEMATICS A PRACTICAL ODYSSEY W/ACCESS

8th Edition

ISBN: 9780357537343

Author: Johnson

Publisher: CENGAGE L

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Chapter 11.5, Problem 5E

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(a)

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The ultimate market shares of the three health clubs, assuming the current trends continue.

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(a) To find: The ultimate market shares of the three health clubs, assuming the current trends continue.

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