4.a) 4. In a simplified version of a classic murder mystery board game, players move between locationsto deduce details about a murder (the murderer, the weapon, and the location). We know thefollowing about the locations and rules for movement:There are six locations: Kitchen, Ballroom, Dining Room, Cellar, Lounge, and Hall.• Dining Room is adjacent to Kitchen, Cellar, and Lounge. Cellar is adjacent to Ballroom and Hall (in addition to the Dining Room)• Kitchen and Ballroom are adjacent (in addition to the above)• Lounge and Hall are adjacent (in addition to the above)• All connections are bi-directional (players can move in both directions)• Players must move to an adjacent location on cach turn and cannot stay in the samelocation• Players move to adjacent locations with equal probabilities at each time stepIf you wish, you can simplify the location using their respective starting letters. I.c. K,B,D,C,L,H(a) Construct the Markov diagram for this the locations of this game and show that theassociate transition matrix T can be written as the following.T=C1I21I2OO000 0-1300-3-300113113300130 0 0001120011200012-INHint: Be sure to state your ordering of the states and justify the entries in T(b) Prove (without computing powers of T) that the matrix T is not a regular matrix.(c) A secret tunnel now added between the Kitchen and Lounge to connect the two areas. Allexisting connections remain and player move with the same rules. L.e. Players cannot stayin the same location and move to adjacent locations with equal probability.i. State (and justify) the new transition matrix R with the new transition probabilitics.ii. Show that this new transition matrix R is regular and determine the stable distributionof this new transition matrix. For this part, you may use a computer program/softwareto assist your calculations. Be sure to state which parts were done using a computerand provide details of the computations you performed.(d) The choice of connecting the Kitchen and Lounge to create a regular transition matrix isnot a unique choice. Starting with the original transition matrix T, provide (and justify)a general criteria required to create a regular transition matrix via the introduction of asingle secret tunnel.

Question

Accepted Answer

From the given information,
The rules are:
There are 6 locations:
Kitchen, Ballroom, Dining Room,…

(a) Construct the Markov diagram for this the locations of this game and show that the associate transition matrix T can be written as the following. T= || 0 1515000 1300-3-3C 0011001 100-KIING 0 0 0 00