LINEAR ALGEBRA
W W L CHEN
c
W W L Chen, 1982, 2005.
This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990.
It is available free to all individuals, on the understanding that it is not to be used for financial gain,
and may be downloaded and/or photocopied, with or without permission from the author.
However, this document may not be kept on any information storage and retrieval system without permission
from the author, unless such system is not accessible to any individuals other than its owners.
Chapter 3
DETERMINANTS
3.1. Introduction
In the last chapter, we have related the question of the invertibility of a square matrix to a question of
solutions of systems of linear equations. In some sense, this is unsatisfactory, since it is not simple to
find an answer to either of these questions without a lot of work. In this chapter, we shall relate these
two questions to the question of the determinant of the matrix in question. As we shall see later, the
task is reduced to checking whether this determinant is zero or nonzero. So what is the determinant?
Let us start with 1
×
1 matrices, of the form
A
= (
a
)
.
Note here that
I
1
= (1). If
a
= 0, then clearly the matrix
A
is invertible, with inverse matrix
A
−
1
= (
a
−
1
)
.
On the other hand, if
a
= 0, then clearly no matrix
B
can satisfy
AB
=
BA
=
I
1
, so that the matrix
A
is not invertible. We therefore conclude that the value
a
is a good “determinant” to determine whether
the 1
×
1 matrix
A
is invertible, since the matrix
A
is invertible if and only if
a
= 0.
Let us then agree on the following definition.
Definition.
Suppose that
A
= (
a
)
is a 1
×
1 matrix. We write
det(
A
) =
a,
and call this the determinant of the matrix
A
.
Chapter 3 : Determinants
page 1 of 23
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Linear Algebra
c
W W L Chen, 1982, 2005
Next, let us turn to 2
×
2 matrices, of the form
A
=
a
b
c
d
.
We shall use elementary row operations to find out when the matrix
A
is invertible. So we consider the
array
(1)
(
A

I
2
) =
a
b
1
0
c
d
0
1
,
and try to use elementary row operations to reduce the left hand half of the array to
I
2
. Suppose first
of all that
a
=
c
= 0. Then the array becomes
0
b
1
0
0
d
0
1
,
and so it is impossible to reduce the left hand half of the array by elementary row operations to the
matrix
I
2
. Consider next the case
a
= 0. Multiplying row 2 of the array (1) by
a
, we obtain
a
b
1
0
ac
ad
0
a
.
Adding
−
c
times row 1 to row 2, we obtain
(2)
a
b
1
0
0
ad
−
bc
−
c
a
.
If
D
=
ad
−
bc
= 0, then this becomes
a
b
1
0
0
0
−
c
a
,
and so it is impossible to reduce the left hand half of the array by elementary row operations to the
matrix
I
2
. On the other hand, if
D
=
ad
−
bc
= 0, then the array (2) can be reduced by elementary row
operations to
1
0
d/D
−
b/D
0
1
−
c/D
a/D
,
so that
A
−
1
=
1
ad
−
bc
d
−
b
−
c
a
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 PETRINA
 Det

Click to edit the document details