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 $M$: