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 definition. The determinant of a 1
×
1
matrix is the entry of the matrix. Once we have defined the determinant of (
n
−
1)
×
(
n
−
1)
matrices, we define 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
) are
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
, j
2
, . . . , j
n
may be called a
permutation
of
{
1
,
2
, . . . , n
}
, 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.
For example, when
n
= 4, the term
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
+
. . .
=
. . .
+ (+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
,
2 is
−
1. [I know this may appear confusing at first glance.] See the
handout Mathematica Session 5.
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
, j
2
, . . . , j
n
, write a sign +1 under
j
i
when
j
i
is in an oddnumbered position when
j
i
, j
i
+1
, . . . j
n
are rearranged in increasing numerical
order, and a sign
−
1 when
j
i
is in an evennumbered position.
Then the sign of the
permutation is the product of the signs under the
j
i
’s. For example, when
j
1
, j
2
, . . . , j
n
=
2
,
6
,
4
,
1
,
5
,
3, we get
2
6
4
1
5
3
−
+
+
+
−
+
,
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.
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 Winter '09
 Wilson
 Linear Algebra, Algebra, Determinant, Matrices, Scalar, Invertible matrix, Det

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