4.2 The Determinant of a Square Matrix
1
Chapter 4.
Determinants
4.2
The Determinant of a Square Matrix
Definition.
The
minor matrix
A
ij
of an
n
×
n
matrix
A
is the (
n
−
1)
×
(
n
−
1) matrix obtained from it by eliminating the
i
th row and the
j
th column.
Example.
Find
A
11
,
A
12
, and
A
13
for
A
=
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
.
Definition.
The determinant of
A
ij
times (
−
1)
i
+
j
is the
cofactor
of
entry
a
ij
in
A
, denoted
A
ij
.
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4.2 The Determinant of a Square Matrix
2
Example.
We can write determinants of 3
×
3 matrices in terms of
cofactors:
det(
A
) =
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
=
a
11

A
11
 −
a
12

A
12

+
a
13

A
13

=
a
11
a
11
+
a
12
a
12
+
a
13
a
13
.
Note.
The following definition is
recursive
.
For example, in order to
process the definition for
n
= 4 you must process the definition for
n
= 3,
n
= 2, and
n
= 1.
Definition 4.1.
The
determinant
of a 1
×
1 matrix is its single entry.
Let
n >
1 and assume the determinants of order less than
n
have been
defined. Let
A
= [
a
ij
] be an
n
×
n
matrix. The
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 Fall '04
 IgorDolgachev
 Determinant, Det, Howard Staunton, Laplace expansion, square matrix, rth row

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