The area of the parallelogram determined by
u
and
v
:
A
u
v
=
O
u
v
·
det
u
v
= det
u
v
Determinants of Order
n
For
A
∈
M
n
×
n
(
F
)
, for
n
≥
2
, denote the
(
n

1)
×
(
n

1)
matrix
obtained from
A
by deleting row
i
and column
j
by
˜
A
ij
.
Definition
Let
A
∈
M
n
×
n
(
F
)
. If
n
= 1
, define
det(
A
) =
A
11
. For
n
≥
2
,
define
det(
A
) =
n
j
=1
(

1)
1+
j
A
1
j
·
det(
˜
A
1
j
)
,
where
det(
A
)
or

A

is the
determinant
of
A
and
c
ij
= (

1)
i
+
j
det(
˜
A
ij
)
is the
cofactor
of
A
ij
.
Note that
det(
A
) =
A
11
c
11
+
A
12
c
12
+
· · ·
+
A
1
n
c
1
n
,
the
cofactor expansion along the first row
of
A
.
Linearity
Theorem 4.3
The determinant of an
n
×
n
matrix is a linear function of each
row when the remaining rows are held fixed:
det
a
1
.
.
.
a
r

1
u
+
kv
a
r
+1
.
.
.
a
n
= det
a
1
.
.
.
a
r

1
u
a
r
+1
.
.
.
a
n
+
k
det
a
1
.
.
.
a
r

1
v
a
r
+1
.
.
.
a
n
for
k
scalar and
u, v, a
i
row vectors in
F
n
.
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 Fall '08
 GUREVITCH
 Linear Algebra, Algebra, Determinant, Det