To determine the matrix $M$, let's analyze the movement of residents among the four regions $A$, $B$, $C$, and $D$, based on the given probabilities:

1. Region A:

   • 55% move to $B$, so the probability of moving from $A$ to $B$ is $0.55$.
   • 15% move to $C$, so the probability of moving from $A$ to $C$ is $0.15$.
   • 25% move to $D$, so the probability of moving from $A$ to $D$ is $0.25$.
   • The probability of staying in $A$ itself is $0$ (since all residents are moving to other regions).

2. Region B:

   • 45% move to $C$, so the probability of moving from $B$ to $C$ is $0.45$.
   • 20% move to $D$, so the probability of moving from $B$ to $D$ is $0.20$.
   • 15% move to $A$, so the probability of moving from $B$ to $A$ is $0.15$.
   • The probability of staying in $B$ itself is $0.20$ (the remaining percentage).

3. Region C:

   • Residents in $C$ either stay in $C$ or move to $A$ with equal probability.
   • This means there’s a 50% probability of staying in $C$, so $P_{CC} = 0.5$.
   • The probability of moving from $C$ to $A$ is also $0.5$, so $P_{CA} = 0.5$.
   • No residents move from $C$ to $B$ or $D$, so $P_{CB} = 0$ and $P_{CD} = 0$.

4. Region D:

   • 65% move to $A$, so $P_{DA} = 0.65$.
   • 15% move to $C$, so $P_{DC} = 0.15$.
   • 10% stay in $D$, so $P_{DD} = 0.10$.
   • The probability of moving from $D$ to $B$ is $0.10$, so $P_{DB} = 0.10$.

Forming the Matrix $M$

The matrix $M$ will be a $4 \times 4$ matrix where each entry $M_{ij}$ represents the probability of moving from region $j$ to region $i$.