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the population in B move to A. Let a, and by denote the respective proportions of the total

population living in A and B at the end of year k. Let us assume that a+b = 1. The

system of linear equations that describe the population proportion at year k+1 is

where A is a transition matrix.

= A

2

a) Compute A for any k integer, and show that

lim A =

3/4

Hint: Use the diagonalization trick to find A-QAQ-¹ where A is the diagonal matrix

formed by the eigenvalues. Then find Aª — QA^Q−¹,

b) Assuming initially a distribution [0o, do]"-". If the migration pattern continues,

compute the long-run population distribution. Will city A be desserted eventually?

Hint: Use this fact:

[C] = []=[] =-

D

e) Implement the solution to 2(a) and 2(b) on Python or MATLAB (with in-built fune-

tions). Submit a screenshot of your solution

Fig: 1