The problem asks us to identify the correct transition matrix that corresponds to the given transition diagram, and to justify our answer. Also, we are asked to explain what a zero represents in a transition matrix. The transition diagram shows the probabilities of switching between magazines A, B, and C from week to week.

1. Problem Description

The problem asks us to identify the correct transition matrix that corresponds to the given transition diagram, and to justify our answer. Also, we are asked to explain what a zero represents in a transition matrix. The transition diagram shows the probabilities of switching between magazines A, B, and C from week to week.

2. Solution Steps

First, we analyze the transition diagram to determine the probabilities. The probabilities on the arrows represent the probability of transitioning from one magazine to another. For example, the probability of staying with magazine A is 55% or 0.

5

5. The probability of transitioning from A to B is 25% or 0.

2

5. The probability of transitioning from A to C is 35% or 0.

3

5. These values make up the first row of the transition matrix. The first row must sum to 1: $0.55 + 0.25 + 0.35 = 1.15$.

Similarly, the probability of transitioning from B to A is 45% or 0.

4

5. The probability of staying with magazine B is 60% or 0.

6

0. The probability of transitioning from B to C is 15% or 0.

1

5. These values make up the second row of the transition matrix. The second row must sum to 1: $0.45 + 0.60 + 0.15 = 1.20$.

Finally, the probability of transitioning from C to A is

0. The probability of transitioning from C to B is 25% or 0.

2

5. The probability of staying with magazine C is 40% or 0.

4

0. These values make up the third row of the transition matrix. The third row must sum to 1: $0.00 + 0.15 + 0.40 = 0.55$.

Now, we can write the transition matrix:

$$$$

\begin{bmatrix}

0.55 & 0.25 & 0.35 \\

0.45 & 0.60 & 0.15 \\

0. & 0.15 & 0.40

\end{bmatrix}

$$$$

Comparing our derived matrix with the given options A, B, C, and D, we see that option D matches our derived matrix.

The elements of the matrix can be explained using the diagram as follows:

- $P(A \rightarrow A) = 0.55$

- $P(A \rightarrow B) = 0.25$

- $P(A \rightarrow C) = 0.35$

- $P(B \rightarrow A) = 0.45$

- $P(B \rightarrow B) = 0.60$

- $P(B \rightarrow C) = 0.15$

- $P(C \rightarrow A) =

0. 0$

- $P(C \rightarrow B) = 0.15$

- $P(C \rightarrow C) = 0.40$

A zero in a transition matrix represents an impossible transition. It means that there is no probability of going from one state to another. In this specific problem, a zero means that no one goes from preferring magazine C to magazine A in a week.

3. Final Answer

Option D is the correct transition matrix. A zero in a transition matrix represents an impossible transition from one state to another.