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 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.
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.
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.
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(i) If the matrix
A
is obtained from
A
by interchanging two rows of
A
, then det(
A
) =
−
det(
A
).
(ii) If the matrix
A
is obtained from
A
by multiplying a row of
A
by a scalar
t
, then
det(
A
) =
t
det(
A
).
(iii) If the matrix
A
is obtained from
A
by adding a scalar multiple of one row of
A
to another, then det(
A
) = det(
A
).
These rules allow the computation of the determinant of a matrix
A
by reducing it to
echelon form while keeping track of how the determinant changes with each row operation.
The proofs below are just sketches. We will fill in details and do examples in class.
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 Winter '08
 Aschbacher
 Linear Algebra, Determinant, Matrices, Scalar, Invertible matrix, Triangular matrix, Det

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