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To solve this problem, we need to construct the matrix \( M \) based on the probabilities of movement between regions \( A \), \( B \), \( C \), and \( D \). This matrix \( M \) will describe how the population in each region changes each year. Here's a step-by-step approach:

### Step 1: Define the matrix structure
Let \( M \) be a \( 4 \times 4 \) matrix, where each entry \( M_{ij} \) represents the fraction of residents moving from region \( j \) to region \( i \).

### Step 2: Assign movement probabilities to the matrix entries
Each column in \( M \) corresponds to a region where people are moving from, and each row corresponds to a region where people are moving to. Let’s analyze the movements:

1. **Residents in \( A \):**
   - 55% move to \( B \).
   - 15% move to \( C \).
   - 25% move to \( D \).
   - The remaining 5% stay in \( A \).

   So, the column for \( A \) in \( M \) is:
   \[
   M_{\cdot A} = \begin{bmatrix} 0.05 \\ 0.55 \\ 0.15 \\ 0.25 \end{bmatrix}
   \]

2. **Residents in \( B \):**
   - 45% move to \( C \).
   - 20% move to \( D \).
   - 15% move to \( A \).
   - The remaining 20% stay in \( B \).

   So, the column for \( B \) in \( M \) is:
   \[
   M_{\cdot B} = \begin{bmatrix} 0.15 \\ 0.20 \\ 0.45 \\ 0.20 \end{bmatrix}
   \]

3. **Residents in \( C \):**
   - Residents stay in \( C \) or move to \( A \) with equal probability, meaning:
     - 50% stay in \( C \).
     - 50% move to \( A \).

   So, the column for \( C \) in \( M \) is:
   \[
   M_{\cdot C} = \begin{bmatrix} 0.50 \\ 0 \\ 0.50 \\ 0 \end{bmatrix}
   \]

4. **Residents in \( D \):**
   - 65% move to \( A \).
   - 15% move to \( C \).
   - 20% stay in \( D \).

   So, the column for \( D \) in \( M \) is:
   \[
   M_{\cdot D} = \begin{bmatrix} 0.65 \\ 0 \\ 0.15 \\ 0.20 \end{bmatrix}
   \]

### Step 3: Construct the matrix \( M \)
Using the above information, we can now write down the full matrix \( M \):

\[
M = \begin{bmatrix} 
0.05 & 0.15 & 0.50 & 0.65 \\ 
0.55 & 0.20 & 0 & 0 \\ 
0.15 & 0.45 & 0.50 & 0.15 \\ 
0.25 & 0.20 & 0 & 0.20 
\end{bmatrix}
\]

### Final Answer
The matrix \( M \) is:
\[
M = \begin{bmatrix} 
0.05 & 0.15 & 0.50 & 0.65 \\ 
0.55 & 0.20 & 0 & 0 \\ 
0.15 & 0.45 & 0.50 & 0.15 \\ 
0.25 & 0.20 & 0 & 0.20 
\end{bmatrix}
\]

Would you like more details on how to interpret or use this matrix?

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Here are 5 related questions to deepen your understanding:

1. What happens to the population distribution over time if we start with an initial distribution?
2. How would we determine the steady-state distribution of the population across regions?
3. How would the matrix change if the movement probabilities were adjusted?
4. What does it mean if all rows of the matrix sum to 1?
5. How could we use this matrix to model changes in population distribution over multiple years?

**Tip:** To determine if a population distribution reaches a stable state, look for an eigenvector of \( M \) with eigenvalue 1, as this represents the steady-state distribution.

A country is divided into four regions (A, B, C, and D). It is found that each year 55% of the residents in A move to B, 15% move to C, and 25% move to D. Of the residents in B, 45% move to C, 20% move to D, and 15% move to A, while the remaining 20% stay in B. The residents of C will either stay where they are or move to A with equal probability. Of the residents in D, 65% move to A, 15% move to C, and 20% stay in D. Model this system with matrix M where x_k+1 = Mx_k.

Learn how to model population movement between regions A, B, C, and D using a transition matrix. This problem applies concepts from probability matrices and Markov chains to predict the population distribution over time. Perfect for college-level students studying population dynamics or matrix algebra.

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