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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

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. Of the residents in D, 65% move to A, 15% move to C, and 15% move to B, while the residents of C will either stay where they are or move to A with equal probability. Determine the matrix M.

This problem involves creating a transition matrix M for population migration across four regions A, B, C, and D. The matrix captures the yearly movement between these regions, with probabilities such as 55% of residents in A moving to B and 65% of residents in D moving to A.

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