This problem involves setting up a transition matrix $M$ for the migration of populations between four regions $A$, $B$, $C$, and $D$. Let's break down the problem and construct the matrix.

We denote the populations in regions $A$, $B$, $C$, and $D$ at time $k$ by the vector: $\mathbf{x}_k = \begin{bmatrix} x_A \\ x_B \\ x_C \\ x_D \end{bmatrix}_k$

The matrix $M$ will describe how the population transitions between these regions each year.

Step-by-Step Construction of $M$

Each entry $M_{ij}$ in matrix $M$ represents the probability of residents moving from region $j$ to region $i$ in one year.

Region A:

• 15% of the residents in $B$ move to $A$.
• 65% of the residents in $D$ move to $A$.
• No one moves from $C$ to $A$ directly.
• 15% of the residents in $C$ stay or move to $A$.

Region