Math 1b Practical — Determinants
February 11, 2009 — Expanded February 18
Square matrices have
determinants
, which are scalars. Determinants can be intro-
duced in several ways; we choose to give a recursive defnition. The determinant o± a 1
×
1
matrix is the entry o± the matrix. Once we have defned the determinant o± (
n
−
1)
×
(
n
−
1)
matrices, we defne the determinant o± an
n
×
n
matrix
A
with entries
a
ij
as
det(
A
)=
a
11
det(
A
11
)
−
a
12
det(
A
12
)+
...
+(
−
1)
n
−
1
a
1
n
det(
A
1
n
)
.
Here
A
ij
denotes the submatrix o±
A
obtained by deleting row
i
and column
j
±rom
A
.
It can be seen inductively that the terms (monomials) that appear in det(
A
)a
r
e
products
a
1
,j
1
a
2
,j
2
···
a
n,j
n
where
j
1
,j
2
,...,j
n
are 1
,
2
,...,n
in some order, each with
a coeﬃcient o± +1 or
−
1. Such a sequence
j
1
2
n
may be called a
permutation
o±
{
1
,
2
}
, and the coeﬃcient o± the term
a
1
,j
1
a
2
,j
2
a
n,j
n
in the determinant expansion
o±
A
is called the
sign
o± the permutation.
For example, when
n
=4,theterm
a
13
a
21
a
34
a
42
arises as
det
⎛
⎜
⎝
a
11
a
12
a
13
a
14
a
21
a
22
a
23
a
24
a
31
a
32
a
33
a
34
a
41
a
42
a
34
a
44
⎞
⎟
⎠
=
+(+1)
a
13
det
⎛
⎝
a
21
a
22
a
24
a
31
a
32
a
34
a
41
a
42
a
44
⎞
⎠
+
=
+ (+1)(+1)
a
13
a
21
det
±
a
32
a
34
a
42
a
44
²
+
=
+ (+1)(+1)(
−
1)
a
13
a
21
a
34
det (
a
42
=
+ (+1)(+1)(
−
1)(+1)
a
13
a
21
a
34
a
42
+
....
So the sign o± 3
,
1
,
4
,
2is
−
1. [I know this may appear con±using at frst glance.] See the
handout Mathematica Session 5.
The ±ollowing rule ±or computing the sign o± a permutation may be extracted ±rom the
method illustrated above. [There are other approaches to understanding signs, and you
may use any o± them.] Given a permutation
j
1
2
n
, write a sign +1 under
j
i
when
j
i
is in an odd-numbered position when
j
i
i
+1
,...j
n
are rearranged in increasing numerical
order, and a sign
−
1wh
en
j
i
is in an even-numbered position.
Then the sign o± the
permutation is the product o± the signs under the
j
i
’s. For example, when
j
1
2
n
=
2
,
6
,
4
,
1
,
5
,
3, we get
±
264153
−
+++
−
+
²
,
so the sign o± the permutation is +. The sign under the 4, ±or example, is + because 4 is
in the third (an odd) position when 4
,
1
,
5
,
3 is reordered as 1
,
3
,
4
,
5.
The ±ollowing rules are extremely important. Some explanation o± why they hold will
be given later, but ±or the moment we just apply them.