Chapter 3.
Determinants.
Let
F
=
R
or
C
and
M
n
the set of all
n
by
n
matrices over
F
. The text deﬁnes
the determinant as a function from
M
n
into
F
satisfying certain axioms. It goes on to
derive various properties of such a function and to show that there exists at most one
such function; ie. it proves a uniqueness result. Finally in section 3.13 it shows such a
function exists, but does not write down the function explicitly.
Our approach will be diﬀerent. We will take the standard approach of deﬁning an
explicit function
det :
M
n
→
F
and proving the function det satisﬁes the axioms in the text and has various other useful
properties. We will not worry about uniqueness.
Permutations.
To deﬁne the determinant we some facts about permutations. Let
I
=
{
1
,... ,n
}
. A
permutation
of
I
is a 11 correspondence
s
:
I
→
I
of
I
with itself. Write
S
n
for the set
of all permutations of
I
. In the literature
S
n
is called the
symmetric group of degree n
.
Cycle Notation.
We need some notation to describe permutations. Each
s
∈
S
n
can
be written in
cycle notation
:
s
= (
a
1
,... ,a
α
)(
b
1
,... ,b
β
)
···
(
z
1
,... ,z
ζ
)
.
In this notation the entries
a
i
,b
i
,... ,z
i
are the members of
I
in some order. The notation
indicates that
s
(
a
i
) =
a
i
+1
for 1
≤
i < α
and
s
(
a
α
) =
a
1
.
Similarly
s
(
z
i
) =
z
i
+1
for
i < ζ
and
s
(
z
ζ
) =
z
1
.
Example 1.
Let
n
= 6. Then
r
= (2
,
1
,
4)(5
,
3
,
6) and
t
= (1)(2
,
6)(3)(4)(5)
are the members of
S
6
such that
2
r
→
1
r
→
4
r
→
2
,
5
r
→
3
r
→
6
r
→
5
and such that
t
ﬁxes 1,3,4, and 5 and interchanges 2 and 6.
1
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The term (
a
1
,... ,a
α
) is called a
cycle
of
s
and this particular cycle is said to be of
length
α
as it involves
α
members of
I
. Thus in Example 1,
r
has two cycles of length 3
and
t
has four cycles of length 1 and one cycle of length 2.
Deﬁne a permutation
s
to be
even
if
s
has an even number of cycles of even length,
and deﬁne
s
to be
odd
if
s
has an odd number of cycles of even length. Thus in Example
1,
r
is even since it has zero cycles of even length, and
t
is odd as it has one cycle of even
length. A permutation like
t
with one cycle of length 2 and
n

2 ﬁxed points is called a
transposition
.
Usually by convention we suppress cycles of length 1. Subject to this convention, we
write the transposition
t
as (2
,
6) rather than (1)(2
,
6)(3)(4)(5).
Deﬁne the
sign function
sgn :
S
n
→ {±
1
}
on
S
n
by sgn(
s
) = +1 if
s
is even and sgn(
s
) =

1 if
s
is odd.
We now deﬁne a function det :
M
n
→
F
called the determinant function. Given a
matrix
A
= (
a
i,j
)
∈
M
n
deﬁne the
determinant
of
A
to be
det(
A
) =
X
s
∈
S
n
sgn(
s
)
·
a
1
,s
(1)
a
2
,s
(2)
···
a
n,s
(
n
)
.
In order to talk about the determinant, we establish some notation: Given
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 Winter '07
 Aschbacher
 Linear Algebra, Algebra, Determinant, Matrices, Det

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