Lesson Element
4.03 b & c Matrices
Instructions and answers for teachers
These instructions cover the student activity section which can be found on page 6. This Lesson
Element supports OCR AS and A Level Further Mathematics A.
When distributing the activity section to the students either as a printed copy or as a Word
file, you will need to remove the teacher instructions section.
Rationale
AS and A level Further Mathematics learners are expected to be able to add, subtract and multiply
conformable matrices; multiply a matrix by a scalar. They must also understand that matrix
multiplication is associative but not commutative. In addition, learners may perform any operations
involving entirely numerical matrices by calculator, including raising square matrices to positive
integer powers. It is therefore important that learners are able to utilise their calculator to perform
these functions.
Assumed knowledge
Learners will need to have been taught the notation and terminology regarding matrices and matrix
algebra. This includes:

The definition of a matrix.

The terms element(s), row(s) and column(s) in relation to a given matrix.

The size of a matrix (m  n) including square matrices, (n  n) .

Understanding when two matrices are equal.
Learners will need to have been taught how and when it is possible, to add, subtract and multiply
matrices. This should also include multiplying every element of a matrix by a scalar quantity.
‘These draft qualifications have not yet been accredited by Ofqual. They are published (along with specimen
assessment materials, summary brochures and sample resources) to enable teachers to have early sight of our
proposed approach.
Further changes may be required and no assurance can be given at this time that the proposed qualifications will be
made available in their current form, or that they will be accredited in time for first teaching in 2017 and first award in
2019 (2018 for AS Level qualifications).’
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Possible misconceptions

Confusing rows and columns in a given matrix.

Any two matrices can be added or subtracted regardless of their relative size.

Matrix multiplication is always commutative, that is, AB = BA.

Matrix multiplication and scalar multiplication are the same.

If A  
 a11 a12 
 b11 b12 
 a11b11 a12b12 
 and B  
 then AB  
.
 a21 a22 
 b21 b22 
 a21b21 a22b22 
Teaching Notes
This task is in two parts. In the first part learners must use their knowledge of matrices to decipher
a number of messages to enable them to access the information of when, where and at what time
two criminal masterminds are meeting to plan their next heist. Learners will be given a set of 10
matrices as part of their ‘code book’ along with instructions on how to translate (into words) the
hidden messages within a given matrix. Any further information required to decipher the messages
will be given within the ‘task information’ or within the task itself. Learners will be given a number of
matrix manipulations to perform which only include matrix addition, subtraction and multiplication.
Learners can be directed by their teacher to work in groups of two or three, with each solving a
particular message or tackling a different part of each message, or the teacher could allow all
learners to work out all three messages separately. In the second part learners must use their
knowledge of matrices to send their own message in response to the messages received by them
in the first part. The second part is more complicated (than the first) and requires learners to have
a deeper understanding of when certain matrices can be added, subtracted and multiplied. This
second part also requires learners to understand and interpret their possible solutions within the
context of the original problem.
Task information
Code Book:
 4

2
A
 5

 6
5

0 4 8
1 2 3 

4 0 2
3
7
 4 2
D

1 1
1 2 3


G  0 1 4
5 6 0


 2 1 0

5 2 4
B
 3 0 0

 6 1 2
 4 14 
E

 3 27 
 3 2 1 


H   6 2 0 
 3 0 2


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3

1
5

4
1 17 1 
 8


17 8 13 18 

C
 6 10 5 12 


 32 0 3 8 
 6 1 
F

 2 15 
 4 14 4 


J   6 8 4 
 64 2 8 


2
 8 13 4 


K   9 7 16 
 8 29
6 

© OCR 2016
Key for translating matrices into ‘words’:
space
a
b
c
d
e
f
g
h
i
j
k
l
0
1
-1
2
-2
3
-3
4
-4
5
-5
6
-6
m
n
o
p
q
r
s
t
u
v
w
x
y
z
7
-7
8
-8
9
-9
10
-10
11
-11
12
-12
13
-13
Interpretation:
To read the hidden message in a matrix the elements should be read row by row i.e. along the first
row followed by the second row, etc. and then translated using the given key. So the matrix below
reveals the message ‘matrices are fun’.
 7 1 10 9 5 2 3 10 


 0 1 9 3 0 3 11 7 
Notation:
D  E means ‘place matrices D and E next to each other to form a single matrix’ so
 4 2 4 14 
DE  

 1 1 3 27 
2 
4


1
1 

D  E means ‘place matrix D above matrix E to form a single matrix’ so D  E 
 4 14 


 3 27 
Answers
Task 1
You are a secret agent working for the government’s intelligence service. You have received three
messages from operatives working undercover giving details of the next meeting of two criminal
masterminds. It is your job to crack the following three messages which will provide some of the
information of where, when and at what time the two criminal masterminds are meeting to plan
their next heist.
Using the ‘task information’ and your knowledge of matrix addition, subtraction and multiplication
determine the information that these three messages contain.
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5 10 13 
 2


0
2
3 7 

Message 1: A + 3B – C =
which gives ‘city centre park’.
 10 9 3
0


 8 1 9 6 
 10

11
Message 2: (ED +2F)  (DE) = 
 10

 1
4 

9 
which gives ‘Thursday’.
2 

13 
3 6 11 3 
 10 12


Message 3: (GH + J)  (HG – K) = 0
10 12 3 7 10  which gives ‘twelve twenty

 13
0 3 5 11 3 

five’.
Where:
city centre park
When:
Thursday
Time:
twelve twenty five
Task 2
Now you know where, when and at what time the two criminal masterminds are meeting you need
to send a message detailing how many undercover agents you are going to send to intercept their
meeting and bring the two criminals into custody. You are concerned, however, that any message
you send could be infiltrated by the criminals and so you decide to send a number of different
messages in which only one of them contains the information of how many undercover agents will
be sent. The messages you send are:
Message A: 32A2 – 2B + 12EF
Message B: GH + 2H + G
Message C: JK – 2DE + 3AD
Message D: 52D2 – 27DE – 183D + 14E
Message E: (DE)F – 11F + E
Message F: DE + 3F + E
Determine which of the six messages contains the information regarding the number of undercover
agents you are going to send – you should give reasons why the other five messages do not
contain the required information. Hence state the number of undercover agents you plan to send.
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Message A does not contain the information because 12EF is a 2 × 2 matrix whilst the other two
(32A2, 2B) are 4 × 4 matrices, so it is not possible to combine them.
 11 4 12 


Message B does not contain the information because GH + 2H + G = 6 5 12 which is


 40 4 9 


‘vgwljw_gq’, which is not a word and -40 does not correspond to a letter.
Message C does not contain the information because it is not possible to calculate AD because A
is a 4 × 4 matrix and D is a 2 × 2 matrix.
 10 12 
 which is
0
 8
Message D does contain the information because 52D2 – 27DE – 183D + 14E = 
‘two’.
 14 15 
 and 14 and 15
 13 2 
Message E does not contain the information because (DE)F – 11F + E = 
do not correspond to letters.
 4 13 
 which is ‘hzpj’,
 8 5 
Message F does not contain the information because DE + 3F + E = 
which is not a word.
In summary:
Message
D
contains the information that
two
undercover agent(s)
should be sent to intercept the criminals and bring them into custody.
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Lesson Element
4.03 b & c Matrices
Student Activity
Task information
Code Book:
 4

2
A
 5

 6
5

0 4 8
1 2 3 

4 0 2
3
 2 1 0

5 2 4
B
 3 0 0

 6 1 2
7
1 17 1 
 8


17 8 13 18 

C
 6 10 5 12 


 32 0 3 8 
 4 14 
E

 3 27 
 4 2
D

1 1
 3 2 1 


H   6 2 0 
 3 0 2


1 2 3


G  0 1 4
5 6 0


3

1
5

4
 6 1 
F

 2 15 
 4 14 4 


J   6 8 4 
 64 2 8 


 8 13 4 


K   9 7 16 
 8 29
6 

Key for translating matrices into ‘words’:
space
a
b
c
d
e
f
g
H
i
j
k
l
0
1
-1
2
-2
3
-3
4
-4
5
-5
6
-6
m
n
o
p
q
r
s
t
u
V
w
x
y
z
7
-7
8
-8
9
-9
10
-10
11
-11
12
-12
13
-13
Interpretation:
To read the hidden message in a matrix the elements should be read row by row i.e. along the first
row followed by the second row, etc. and then translated using the given key. So the matrix below
reveals the message ‘matrices are fun’.
 7 1 10 9 5 2 3 10 


 0 1 9 3 0 3 11 7 
Version 1
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6
© OCR 2016
Notation:
D  E means ‘place matrices D and E next to each other to form a single matrix’ so
 4 2 4 14 
DE  

 1 1 3 27 
2 
4


1
1 
D  E means ‘place matrix D above matrix E to form a single matrix’ so D  E  
 4 14 


 3 27 
Task 1
You are a secret agent working for the government’s intelligence service. You have received three
messages from operatives working undercover giving details of the next meeting of two criminal
masterminds. It is your job to crack the following three messages which will provide some of the
information of where, when and at what time the two criminal masterminds are meeting to plan their
next heist.
Using the ‘task information’ and your knowledge of matrix addition, subtraction and multiplication
determine the information that these three messages contain.
Message 1: A + 3B – C
Message 2: (ED +2F)  (DE)
Message 3: (GH + J)  (HG – K)
Where:
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When:
Time:
7
© OCR 2016
Task 2
Now you know where, when and at what time the two criminal masterminds are meeting you need to
send a message detailing how many undercover agents you are going to send to intercept their
meeting and bring the two criminals into custody. You are concerned, however, that any message
you send could be infiltrated by the criminals and so you decide to send a number of different
messages in which only one of them contains the information of how many undercover agents will
be sent. The messages you send are:
Message A: 32A2 – 2B + 12EF
Message B: GH + 2H + G
Message C: JK – 2DE + 3AD
Message D: 52D2 – 27DE – 183D + 14E
Message E: (DE)F – 11F + E
Message F: DE + 3F + E
Determine which of the six messages contains the information regarding the number of undercover
agents you are going to send – you should give reasons why the other five messages do not contain
the required information. Hence state the number of undercover agents you plan to send.
Message A does / does not contain the information because
Message B does / does not contain the information because
Message C does / does not contain the information because
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Message D does / does not contain the information because
Message E does / does not contain the information because
Message F does / does not contain the information because
In summary:
Message
contains the information that
undercover agent(s)
should be sent to intercept the criminals and bring them into custody.
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© OCR 2016