ELEMENTARY MATHEMATICS
W W L CHEN and X T DUONG
c
W W L Chen, X T Duong and Macquarie University, 1999.
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Chapter 2
INTRODUCTION TO MATRICES
2.1.
Linear Equations
Example 2.1.1.
Consider the two linear equations
3
x
+ 4
y
= 11
,
5
x
+ 7
y
= 19
.
It is easy to check the two equations are satisfied when
x
= 1 and
y
= 2. We can represent these two
linear equations in matrix form as
3
4
5
7
x
y
=
11
19
,
where we adopt the convention that
3
4
•
•
x
y
=
11
•
and
•
•
5
7
x
y
=
•
19
represent respectively the information 3
x
+ 4
y
= 11 and 5
x
+ 7
y
= 19. Under this convention, it is easy
to see that
1
0
0
1
x
y
=
x
y
for every
x, y
∈
R
. Next, observe that
7
−
4
−
5
3
3
4
5
7
=
1
0
0
1
,
†
This chapter was written at Macquarie University in 1999.
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W W L Chen and X T Duong : Elementary Mathematics
where, under a convention slightly more general to the one used earlier, we have
7
−
4
•
•
3
•
5
•
=
1
•
•
•
,
7
−
4
•
•
•
4
•
7
=
•
0
•
•
,
•
•
−
5
3
3
•
5
•
=
•
•
0
•
,
•
•
−
5
3
•
4
•
7
=
•
•
•
1
,
representing respectively (7
×
3)+((
−
4)
×
5) = 1, (7
×
4)+((
−
4)
×
7) = 0, ((
−
5)
×
3)+(3
×
5) = 0 and
((
−
5)
×
4) + (3
×
7) = 1. It now follows on the one hand that
7
−
4
−
5
3
3
4
5
7
x
y
=
1
0
0
1
x
y
=
x
y
,
and on the other hand that
7
−
4
−
5
3
3
4
5
7
x
y
=
7
−
4
−
5
3
11
19
=
1
2
.
The convention mentioned in the example above is simply the rule concerning the multiplication of
matrices. The purpose of this chapter is to study the arithmetic in connection with matrices. We shall
be concerned primarily with 2
×
2 real matrices. These are arrays of real numbers of the form
a
11
a
12
a
21
a
22
,
consisting of two rows counted from top to bottom, and two columns counted from left to right. An
entry
a
ij
thus corresponds to the entry in row
i
and column
j
.
2.2.
Arithmetic
ADDITION AND SUBTRACTION.
Suppose that
A
=
a
11
a
12
a
21
a
22
and
B
=
b
11
b
12
b
21
b
22
are two
2
×
2
matrices. Then
A
+
B
=
a
11
+
b
11
a
12
+
b
12
a
21
+
b
21
a
22
+
b
22
and
A
−
B
=
a
11
−
b
11
a
12
−
b
12
a
21
−
b
21
a
22
−
b
22
.
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