Application of Linear Algebra: The city of Nabal has four political parties: The

Question

Application of Linear Algebra:

The city of Nabal has four political parties: The city of Nabal has four political parties. Anarchists, Democrats, Libertarians, and Republicans. The following table shows the transition probabilities between the political parties, the top row being the initial political party and the side row being the party of the next year. Approximately how many members will each party have after 20 years if the adult population of Nabal is 30,000?  

A D L R A 1/10 1/4 1/10 1/5 D 3/10 1/4 1/10 3/10 L 3/10 1/4 7/10 1/5 R 3/10 1/4 1/10 3/10

Explanation / Answer

Let M =

1/10

1/4

1/10

1/5

3/10

1/4

1/10

3/10

3/10

1/4

7/10

1/5

3/10

1/4

1/10

3/10

Then the steady state vector is given by MX = X or, (M-I4)X = 0. To solve this, equation, we will reduce to its RREF,as under, the matrix M-I4=

-9/10

1/4

1/10

1/5

3/10

-3/4

1/10

3/10

3/10

1/4

-3/10

1/5

3/10

1/4

1/10

-7/10

Multiply the 1st row by -10/9

Add -3/10 times the 1st row to the 2nd row

Add -3/10 times the 1st row to the 3rd row

Add -3/10 times the 1st row to the 4th row

Multiply the 2nd row by -3/2

Add -1/3 times the 2nd row to the 3rd row

Add -1/3 times the 2nd row to the 4th row

Multiply the 3rd row by -5

Add -1/5 times the 3rd row to the 4th row

Add 1/5 times the 3rd row to the 2nd row

Add 1/9 times the 3rd row to the 1st row

Add 5/18 times the 2nd row to the 1st row

Then the RREF of M-I4 is

1

0

0

-3/4

0

1

0

-1

0

0

1

-9/4

0

0

0

0

Thus, if X = (x,y,z,w)T, the equation (M-I4)X = 0 is equivalent to

x-3w/4 = 0 or, x = 3w/4, y-w = 0 or, y = w, and z-9w/4 = 0 or, z = 9w/4. Hence X = (3w/4,w,9w/4,w)T = w(3/4,1,9/4,1)T. Thus, the steady state vector is(3/4,1,9/4,1)T. The number of members in the Anarchist, Democrats, Libertarian, and Republican parties after 20 years will be (3/20)*30000, = 4500, (1/5)*30000= 6000, (9/20)*30000= 13500, and (1/5)*30000= 6000 respectively.

Note: (3/4)+1+(9/4)+1 =5. Hence each of 3/4,1,9/4,1 has been divided by 5, before multiplying by 30000, to compute the number of party members.

1/10

1/4

1/10

1/5

3/10

1/4

1/10

3/10

3/10

1/4

7/10

1/5

3/10

1/4

1/10

3/10

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