The problem describes a scenario where a new arcade, "Amazing Arcade" (A), is opening near "Extra Emporium" (E). The management of Extra Emporium is using a matrix model to predict shopper behavior. The state matrix $S_n$ represents the number of shoppers at each location after $n$ weeks. The recurrence relation is given by $S_{n+1} = T S_n$, where $T$ is the transition matrix and $S_0$ is the initial state matrix. The transition matrix $T$ is given as $T = \begin{bmatrix} 0.72 & 0.66 \\ 0.28 & 0.34 \end{bmatrix}$ and the initial state is $S_0 = \begin{bmatrix} 150000 \\ 80000 \end{bmatrix}$. We are asked to (a) draw a transition diagram, (b) explain the meaning of element $T_{12}$, and (c) explain what it means that the columns of the transition matrix add to 1.

1. Problem Description

The problem describes a scenario where a new arcade, "Amazing Arcade" (A), is opening near "Extra Emporium" (E). The management of Extra Emporium is using a matrix model to predict shopper behavior. The state matrix $S_n$ represents the number of shoppers at each location after $n$ weeks. The recurrence relation is given by $S_{n+1} = T S_n$, where $T$ is the transition matrix and $S_0$ is the initial state matrix. The transition matrix $T$ is given as

$T = \begin{bmatrix} 0.72 & 0.66 \\ 0.28 & 0.34 \end{bmatrix}$ and the initial state is $S_0 = \begin{bmatrix} 150000 \\ 80000 \end{bmatrix}$. We are asked to (a) draw a transition diagram, (b) explain the meaning of element $T_{12}$, and (c) explain what it means that the columns of the transition matrix add to

1.

2. Solution Steps

(a) Transition Diagram:

The transition diagram represents the movement of shoppers between Amazing Arcade (A) and Extra Emporium (E).

- From A to A: 72% (0.72) of shoppers at A stay at A the next week.

- From A to E: 28% (0.28) of shoppers at A move to E the next week.

- From E to A: 66% (0.66) of shoppers at E move to A the next week.

- From E to E: 34% (0.34) of shoppers at E stay at E the next week.

The transition diagram will have two states, A and E. Arrows will connect the states, with percentages indicating the transition probabilities.

(b) Meaning of $T_{12}$:

$T_{12}$ is the element in the first row and second column of the transition matrix $T$. In this case, $T_{12} = 0.66$. This represents the proportion of shoppers at Extra Emporium (E) this week that will be at Amazing Arcade (A) next week. Specifically, 66% of the shoppers at Extra Emporium this week will be at Amazing Arcade next week.

(c) Columns adding to 1:

The columns of the transition matrix adding to 1 means that the total number of shoppers remains constant. In other words, no shoppers are entering or leaving the system (the combination of Amazing Arcade and Extra Emporium). All shoppers from one week are accounted for in the next week, either staying at the same location or switching to the other location. For example, 0.72 + 0.28 = 1 for the first column, and 0.66 + 0.34 = 1 for the second column. This indicates that all shoppers from A either stay at A or move to E, and all shoppers from E either move to A or stay at E.

3. Final Answer

(a) Transition Diagram:

(A) <--(0.72) (A), (A) -->(0.28) (E)

(E) <--(0.34) (E), (E) -->(0.66) (A)

(b) The element $T_{12} = 0.66$ represents the proportion (or percentage) of shoppers at Extra Emporium (E) this week that will be at Amazing Arcade (A) next week. 66% of shoppers at E will be at A next week.

(c) The columns of the transition matrix adding to 1 means that the total number of shoppers remains constant. No shoppers are entering or leaving the system.