a Organise the information given on the diagram in a table.
B
b Use the transition matrix from Part A.
C
c Use matrix multiplication.
A
aTransition matrix:
0.35& 0.12 0.65& 0.88
B
b 20 218
C
c 4357
Practice makes perfect
a Let's use a table to organize the information given in the question about the change in player ownership.
From
CD
DAP
To
CD
35 %
12 %
DAP
65 %
88 %
We can write a transition matrix corresponding to this table.
Let's use probabilities instead of percentages in the matrix.
0.35& 0.12 0.65& 0.88
b Let's see why the transition matrix of Part A is useful. To do this, let's use a 2* 1 matrix for the number of people in Central City who own a portable CD player (c) and a digital audio player (d) at any given point.
c s
Let's calculate and interpret the product matrix.
0.35& 0.12 0.65& 0.88
c d =
0.35c+0.12d 0.65c+0.88dThe result is a 2* 1 matrix, and the elements are the sum of two products.
The expression 0.35c gives the number of people who keep their CD players.
The expression 0.12d gives the number of people who switch from using digital audio player to a CD player.
The sum 0.35c+0.12d is therefore the number of people in Central City who will use CD players next year. This is the first element of the product matrix.
Similarly, 0.65c+0.88d is the number of people in Central City who will use digital audio players next year. This is the second element of the product matrix.
In this part of the question, we are given the number of people who currently use CD players and digital audio players.
c&=7748 d&=17 252
We are asked to predict the player ownership next year.
According to our interpretation of the product above, this can be done by calculating the following product matrix.
0.35& 0.12 0.65& 0.88
7748 17 252
Let's evaluate the product.
Since the second element of the result corresponds to the people who use digital audio players, according to this trend there will be about 20 218 people in Central City who use digital audio players next year.
c According to the interpretation of the matrix product in Part B, we can find the population in the previous year by solving the following matrix equation.
0.35& 0.12 0.65& 0.88
c d =
7748 17 252
A matrix equation like this can be solved using the inverse matrix.
MX=P ⟹ X=M^(-1)P
Use the determinant to find the inverse of a 2* 2 matrix.
M= 0.35& 0.12 0.65& 0.88
⇓
|M|=0.35(0.88)-0.12(0.65)=0.23
⇓
M^(-1)=1/0.23 0.88& -0.12 -0.65& 0.35
Let's use this inverse to find the answer to the question.
Since the second element of the result corresponds to the people who use digital audio players, according to this trend there were about 4357 people in Central City who used digital audio players last year.
