Math 1b Practical — Determinants
January 30, 2011
Square matrices have
determinants
, which are scalars. Determinants can be intro-
duced in several ways; we choose to give a recursive defnition. The determinant oF a 1
×
1
matrix is the entry oF the matrix. Once we have defned the determinant oF (
n
−
1)
×
(
n
−
1)
matrices, we defne the determinant oF 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 oF
A
obtained by deleting row
i
and column
j
From
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 oF +1 or
−
1. Such a sequence
j
1
2
n
may be called a
permutation
oF
{
1
,
2
}
, and the coeﬃcient oF the term
a
1
,j
1
a
2
,j
2
a
n,j
n
in the determinant expansion
oF
A
is called the
sign
oF the permutation.
±or 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 oF 3
,
1
,
4
,
2is
−
1.
The Following rule For computing the sign oF a permutation may be extracted From the
method illustrated above. [There are other approaches to understanding signs, and you
may use any oF 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 oF the
permutation is the product oF the signs under the
j
i
’s. ±or example, when
j
1
2
n
=
2
,
6
,
4
,
1
,
5
,
3, we get
±
264153
−
+++
−
+
²
,
so the sign oF the permutation is +. The sign under the 4, For example, is + because 4 is
in the third (an odd) position when 4
,
1
,
5
,
3 is reordered as 1
,
3
,
4
,
5.
The Following rules are extremely important. Some explanation oF why they hold will
be given later, but For the moment we just apply them.
(o) The determinant oF the identity matrix is 1.