Graphics Reference
In-Depth Information
The above Laplace expansion can be used for any order determinant, and for
the purposes of this topic, the highest order we will encounter is a
fourth-order
determinant
. So let's expand the following determinant
04
−
10
10
14
det
A
=
.
06
33
35
22
The first column and the first row both contain two zeros, which helps our expansion,
so let's expand
|
A
|
using the first row. The two relevant minor determinants are
114
033
322
104
063
352
A
12
=
,
13
=
.
Next, we expand
A
12
and
A
13
with their cofactors as follows
4
1
(
1
)
1
+
1
+
1
)
1
+
2
+
1
)
1
+
3
33
22
03
32
03
32
c
12
=−
−
1
(
−
4
(
−
=−
4 [0
+
9
−
36]
=
108
1
1
(
1
)
1
+
1
+
1
)
1
+
3
63
52
06
35
c
13
=−
−
4
(
−
=−
1 [
−
3
−
72]
=
75
.
Therefore,
det
A
=
c
12
+
c
13
=
108
+
75
=
183
.
4.10 Cofactors and Inverse of a Matrix
Although the idea of cofactors has been described in the context of determinants,
they can also be applied to matrices. For example, let's start with the following
matrix and its cofactor matrix
⎡
⎤
013
214
426
⎣
⎦
A
=
⎡
⎤
A
11
A
12
A
13
⎣
⎦
cofactor matrix of
A
=
A
21
A
22
A
23
A
31
A
32
A
33
