Answer:
(a)
![\mathbf{P_2 = \left[\begin{array}{c}140 \\ 160 \\ 200 \end{array}\right]}](https://img.qammunity.org/2021/formulas/mathematics/college/j5mjvwe3u24osis6ku03w0orpd8llnfspk.png)
(b) After an infinite period of time; we will get back to a result similar to after the two time period which will be
![= \left[\begin{array}{c}140 \\ 160 \\ 200 \end{array}\right]}](https://img.qammunity.org/2021/formulas/mathematics/college/9v008cncvbb47y8nq2nw5bklazag9o2jhw.png)
Explanation:
The Markov Matrix can be interpret as :
![M = \left[\begin{array}{ccc} (1)/(5) & (2)/(5) &(1)/(5) \\ \\ (2)/(5)&(2)/(5)&(1)/(5)\\ \\ (2)/(5)& (2)/(5)& (2)/(5) \end{array}\right]](https://img.qammunity.org/2021/formulas/mathematics/college/i3bmfdqm610larj008yy6h2j3p8e8t31dg.png)
From (a) ; we see that the initial population are as follows: 130 individuals in location 1, 300 in location 2, and 70 in location 3.
Le P represent the Population; So ;
![P = \left[\begin{array}{c}130 \\ 300 \\ 70 \end{array}\right]](https://img.qammunity.org/2021/formulas/mathematics/college/ca35gry2u9d8siohgcyz9kl3j738w1jdrx.png)
The objective is to find How many are in each location after two time periods;
So, after two time period ; we have the population
![P_2 = [M]^2 [P]](https://img.qammunity.org/2021/formulas/mathematics/college/93dsv9zm81m2wqqf9n9bzfqtqix1ngs908.png)
where;
![[M]^ 2 = \left[\begin{array}{ccc} (1)/(5) & (2)/(5) &(1)/(5) \\ \\ (2)/(5)&(2)/(5)&(1)/(5)\\ \\ (2)/(5)& (2)/(5)& (2)/(5) \end{array}\right] \left[\begin{array}{ccc} (1)/(5) & (2)/(5) &(1)/(5) \\ \\ (2)/(5)&(2)/(5)&(1)/(5)\\ \\ (2)/(5)& (2)/(5)& (2)/(5) \end{array}\right]](https://img.qammunity.org/2021/formulas/mathematics/college/dmiv37pbagrsp7wg2zr1kuhl9ld7iz64z5.png)
![[M]^ 2 = (1)/(25) \left[\begin{array}{ccc} 1+2+4 & 1+2+4 &1+2+4 \\ \\ 2+2+4&2+2+4&2+2+4\\ \\ 2+4+4&2+4+4& 2+4+4 \end{array}\right]](https://img.qammunity.org/2021/formulas/mathematics/college/vjuz165q505qly5j109cu0vo9qw8nr9zh3.png)
![[M]^ 2 = (1)/(25) \left[\begin{array}{ccc}7&7&7 \\ \\ 8 &8&8\\ \\10&10& 10 \end{array}\right]](https://img.qammunity.org/2021/formulas/mathematics/college/eohmj9rgken7mw6v75gd3lr5oz6qtuak4n.png)
Now; Over to after two time period ; when the population
![P_2 = (1)/(25) \left[\begin{array}{ccc}7&7&7 \\ \\ 8 &8&8\\ \\10&10& 10 \end{array}\right] \left[\begin{array}{c}130 \\ 300 \\ 70 \end{array}\right]](https://img.qammunity.org/2021/formulas/mathematics/college/2hv71znl8a0ihoek08n1hdlgdhh96ucccy.png)
![\mathbf{P_2 = \left[\begin{array}{c}140 \\ 160 \\ 200 \end{array}\right]}](https://img.qammunity.org/2021/formulas/mathematics/college/j5mjvwe3u24osis6ku03w0orpd8llnfspk.png)
(b) The total number of individuals in the migration process is 500. After a long time, how many are in each location?
After a long time; that is referring to an infinite time (n)
So;
![P_n = [M]^n [P]](https://img.qammunity.org/2021/formulas/mathematics/college/a1pe89yzwfwdvvbzhvy1eq3oq5jqmatumj.png)
where ;
![[M]^n \ can \ be \ [M]^2 , [M]^3 , [M]^4 .... \infty](https://img.qammunity.org/2021/formulas/mathematics/college/52npinxdibekctd3tyhytvkrl0980fvwqi.png)
; if we determine the respective values of
we will always result to the value for
; Now if
is said to be a positive integer; then :
After an infinite period of time; we will get back to a result similar to after the two time period which will be
![= \left[\begin{array}{c}140 \\ 160 \\ 200 \end{array}\right]}](https://img.qammunity.org/2021/formulas/mathematics/college/9v008cncvbb47y8nq2nw5bklazag9o2jhw.png)