T= 00 3 w|wo|wo|TOO 3 1|21|20 H2H2OOO 1|21|200 00 00 0 0 Hint: Be sure to state your ordering of the states and justify the entries in T Show that X = 1 is an eigenvalue of T
T= 00 3 w|wo|wo|TOO 3 1|21|20 H2H2OOO 1|21|200 00 00 0 0 Hint: Be sure to state your ordering of the states and justify the entries in T Show that X = 1 is an eigenvalue of T
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 20T
Related questions
Question
Construct the Markov diagram for this game and show that the associate transition matrix T can be written as the following. Show that λ = 1 is an eigenvalue of T. If can answer both (a)and(b), else answer (b).
![In a simplified version of a popular online game, a team of Champions move between locations
to protect their own Nexus, whilst attempting to destroy the opponent team's Nexus. We know
the following about the locations and rules for movement:
• A team consists of 5 Champions
• There are five locations: Home (Nexus),Enemy (Nexus), Safe lane, Mid lane, and Off lane.
• Home is adjacent to the Safe lane, Mid lane, and Off lane
• Enemy is adjacent to the Safe lane, Mid lane, and Off lane
• No other connections exist
• All connections are bi-directional (Champions can move in both directions)
• For safety reasons, Champions must move to an adjacent location at each time step and
cannot stay in the same location
• Champions move to adjacent locations with equal probabilities at each time step and
cannot stay in the same location.
If you wish, you can simplify the location using their respective starting letters. I.e. H,E,S,M,0.
Stylistic note: If you have a strong preference, you may use Ancient/Hero in place of Nexus/Champion.
(a) Construct the Markov diagram for this game and show that the associate transition matrix
T can be written as the following.
1
1
T =
0 0
Hint: Be sure to state your ordering of the states and justify the entries in T
(b) Show that A= 1 is an eigenvalue of T
HINHIN O O
O O H1 313
O O 1/3H3113](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8f86f970-1556-4001-95b2-53eb6820da66%2F24baa697-d0c3-4c66-8009-399bd44b308a%2F3m1buehp_processed.png&w=3840&q=75)
Transcribed Image Text:In a simplified version of a popular online game, a team of Champions move between locations
to protect their own Nexus, whilst attempting to destroy the opponent team's Nexus. We know
the following about the locations and rules for movement:
• A team consists of 5 Champions
• There are five locations: Home (Nexus),Enemy (Nexus), Safe lane, Mid lane, and Off lane.
• Home is adjacent to the Safe lane, Mid lane, and Off lane
• Enemy is adjacent to the Safe lane, Mid lane, and Off lane
• No other connections exist
• All connections are bi-directional (Champions can move in both directions)
• For safety reasons, Champions must move to an adjacent location at each time step and
cannot stay in the same location
• Champions move to adjacent locations with equal probabilities at each time step and
cannot stay in the same location.
If you wish, you can simplify the location using their respective starting letters. I.e. H,E,S,M,0.
Stylistic note: If you have a strong preference, you may use Ancient/Hero in place of Nexus/Champion.
(a) Construct the Markov diagram for this game and show that the associate transition matrix
T can be written as the following.
1
1
T =
0 0
Hint: Be sure to state your ordering of the states and justify the entries in T
(b) Show that A= 1 is an eigenvalue of T
HINHIN O O
O O H1 313
O O 1/3H3113
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