Chapter 2.1, Problem 2P

Explanation of Solution

Relating two vectors using matrix multiplication:

Suppose that at the beginning of each year, people change the beer they drink.

Let for $i=1,2,3,\mathrm{....}{x}_i$ be the number of people who prefer beer “i” at the beginning of this month, and $y_i$ be the number of people who prefer beer “i” at the beginning of next month.

Since 30% of the people who prefer beer 1 switch to beer 2 and 20% of the people switch to beer 3, so the remaining 50% continue to be with beer 1. Also, 10% of people who prefer beer 3 switch to beer 1.

Hence, the number of people who prefer beer 1 in the beginning of next month is given below:

  $y_1=0.5x_1+0.1x_3$

Now, 30% of the people who prefer beer 2 switch to beer 3 and the remaining 70% of the people continue to be with beer 2. Also, 30% of people who prefer beer 3 switch to beer 2 and 30% of the people who prefer beer 1 switch to beer 2.

Hence, the number of people who prefer beer 2 in the beginning of next month is given below:

  $y_2=0.3x_1+0.7x_2+0.3x_3$

Then, 30% of the people who prefer beer 3 switch to beer 2 and 10% of the people switch to beer 1, so the remaining 60% continue to be with beer 3. Also, 20% of people who prefer beer 1 switch to beer 3 and 30% of the people who prefer beer 2 switch to beer 3.

Hence, the number of people who prefer beer 3 in the beginning of next month is given below:

  $y_3=0.2x_1+0$