VCE General Maths (Further) Unit 1
Matrices
2016 Test
Name: _______________________________
Total:
Conditions:
/
39 marks
(
%)
20 minutes + 20 minutes = 40 minutes, Casio Calculator and Summary Book
Calculator-Free Section
20 minutes
Question 1
Define the order of a matrix:
__________________________________________________________________________________
__________________________________________________________________________________
1 mark
The following matrices will be used for Questions 2, 3 & 4
1 2
𝐴 = [0 3
1 0
1
−2]
2
3 2 3
𝐵 = [ 3 1 0]
3 0 6
1 1
𝐶 = [2 1
0 0
0
−2]
1
Question 2
Element b2,3 is ______________
1 Mark
Question 3
𝐴+𝐶 =
1 Mark
Question 4
3𝐵 =
1 Mark
Question 5
Solve the following matrix equation for 𝑥, 𝑦 𝑎𝑛𝑑 𝑧.
[
𝟏 𝒙
𝟑 𝟒
𝟒 𝟓
]+[
]=[
]
𝒚 𝟐
𝟓 𝒛
𝟑 −𝟐
2 Marks
Question 6
Five trout-breeding ponds, P, Q, R, X and V, are connected by pipes, as shown in the diagram below:
P
Q
V
R
V
X
X
Matrix W is an adjacency matrix used to represent the information in the diagram.
0
1
𝑊= 1
1
[0
1
0
0
1
0
P
1
0
0
1
0
Q
1
1
1
0
1
R X V
0
0
0
1
0]
P
Q
R
X
V
a) Find the sum of the elements in row 3 of matrix W.
____________________________________________________________________________
1 Mark
b) In terms of the breeding ponds, what does the sum of the elements in row 3 of matrix W
represent?
____________________________________________________________________________
1 Mark
Question 7
Given the following transition diagram, complete the transition table.
80%
20%
45%
B
A
55%
From
From
A
To
B
A
B
2 Marks
Question 8
Construct a (3 x 3) identity matrix.
1 Mark
The following matrices will be used for questions 9 – 14
𝟒 𝟐
𝑫=[
]
𝟔 𝟑
𝟑
𝑬=[ ]
𝟓
𝟐
𝟒
𝑭=[
]
−𝟐 −𝟑
𝟐
𝑮 = [𝟑]
𝟒
𝑯 = [𝟏 𝟐 𝟑]
Question 9
Can 𝐸𝐹 be defined? Justify your answer.
1 Mark
Question 10
Define the order of the solution of 𝐺𝐻
1 Mark
Question 11
Calculate 𝐷𝐹
3 Marks
Question 12
Calculate the determinant of Matrix F
1 Mark
Question 13
Calculate the inverse of Matrix 𝐹
1 Mark
Question 14
Which of the matrices (D, E or F) is a singular matrix? Justify your answer.
2 Marks
Calculator-Active Section
20 minutes
Question 1
4
[
6
23
] =
3
1 Mark
Question 2
𝑥
If [
𝑧
𝑦 1 −1
3 5
][
]=[
]
𝑣 4 3
−2 4
Find (in fraction form) [
𝑥
𝑧
𝑦
]=
𝑣
1 Mark
Question 2
Fifty students from CCB travel from Derrin to Bower. In the first class carriage tickets cost $25, in the
second class carriage tickets cost $20.The total ticket sales for the trip cost $1200.
Find the number of tickets of each type sold by following the steps below, where f is the number of
first class tickets sold and s is the number of second class tickets sold.
a) Fill in the missing values in the equation:
𝑓 + 𝑠 = 50
25𝑓 +
𝑠=
1 Mark
b) Construct a matrix equation to represent the simultaneous equations.
3 Marks
c) Using the inverse matrix, solve the matrix equation and hence state the number of each
type of ticket sold.
3 Marks
Question 3
For students living in Bendigo, there are three main ways of getting to school; riding, walking and
taking the bus. Students change their mode of transport each week as indicated by the transition
matrix below.
There are 250 students in year 7 at Catholic College Bendigo. Initially, 80 students ride, 20 students
walk and 150 students take the bus.
this week
ride
walk
bus
ride
walk
next week
bus
a) Complete the transition diagram below to represent this data. You will need to add
percentages and arrows to the three unmarked bold connections below.
40%
Bus
20%
10%
40%
Walk
Ride
20%
80%
3 Marks
b) What percentage of students who walked to school this week will catch the bus to school
next week?
1 Mark
c) Prepare an initial state matrix.
1 mark
d) Write an appropriate equation and then calculate how many students will ride next week?
3 marks
e) Show that in the long term, 125 students walk to school?
2 marks
End of assessment task