Maths Quest Maths C Year 12 for Queensland Chapter 2 Matrices and applications WorkSHEET 2.2 2

WorkSHEET 2.2 Matrices and applications Name: ______

Questions 1 to 3 refer to the following information.
In the local cricket competition teams can use either of two types of ball — Kingfisher or Best Match. At the end of each season clubs sometimes decide to change the ball they use. Research suggests 80% of those using Kingfisher stay with that ball for the next season and 70% of those using Best Match continue to use that ball. At the beginning of this season 80% of clubs used Best Match and the remainder used Kingfisher.
1  / Write the transition matrix for this system. / From
To
The transition matrix is M =
2  / What proportion of clubs will be using Kingfisher in 3 years time? /
55% of clubs will be using Kingfisher balls in 3years’ time.
3  / What is the long-term behaviour of the system? / The long-term behaviour of the system can be found by calculating

on your graphics calculator.
The long-term behaviour of the system is
.
4  / Calculate the eigenvalues of the matrix / The eigenvalues are those values, l, for which
That is

Questions 5, 6 and 7 relate to the following information.
Consider the following two-industry economy consisting of raw materials (R) and manufactured goods (M). To produce 1 unit of R requires the input of 0.2 units of R and 0.4units of M. To produce 1 unit of M requires the input of 0.5 units of R and 0.3 units of M.
5  / Write the consumption matrix for this economy. / The consumption matrix, C, is given by

or

6  / What input is required to produce 4 units of R and 3 units of M? /
The input required is 2.3 units of R and 2.5units of M.
7  / What production is needed to meet an external demand of 12 units of R and 25 units of M? /
You need to produce 58.1 units of R and 68.9units of M.
Questions 8, 9 and 10 refer to the following information.
The Leslie matrix describing the yearly changes in the number of female animals in a certain population is given below:

8  / What does the number 2.5 in the first row of the matrix indicate? / This number indicates that each two-year-old female produces on average 2.5 offspring per year.
9  / What does the number 0.3 in the second row of the matrix represent? / This number indicates that 30% of 1-year-old animals survive to two years of age.
10  / If the initial population was
Number of 1-year-olds = 220
Number of 2-year-olds = 180
Number of 3-year-olds = 260
calculate the population in 3 years time. / The population in 3 years time, N3, is given by:

The population in 3 years time will be
Number of 1-year-olds = 793.5
Number of 2-year-olds = 103.5
Number of 3-year-olds = 145.5