Game Development Reference
In-Depth Information
Using cofactors, a method for calculating the determinant of an nn
matrix
×
can be expressed as follows. First, define the determinant of a 11
×
matrix to be
the entry of the matrix itself. Then the determinant of an nn
matrix M is given
×
by both the formula
n
()
det
M
=
MC
M
(3.54)
ik
ik
i
=
1
and the formula
n
()
det
M
=
MC
M ,
(3.55)
kj
kj
j
=
1
where k is an arbitrarily chosen constant such that 1
. Remarkably, both
formulas give the same value for the determinant regardless of the choice of k .
The determinant of M is given by the sum along any row or column of products
of entries of M and their cofactors.
≤≤
kn
An explicit formula for the determinant of a 22
×
matrix is easy to extract
from Equations (3.54) and (3.55):
cd =−
bc
.
(3.56)
We also give an explicit formula for the determinant of a 3
×
3
matrix. The fol-
lowing is written as one would evaluate Equation (3.55) with
=
1
.
k
aaa
11
12
13
aa
aa
aa
22
23
21
23
21
22
aaa a
=
a
+
a
21
22
23
11
12
13
aa
aa
aa
32
33
31
33
31
32
aaa
31
32
33
(
)
(
)
=
aaa aa aaa aa
aaa aa
11
22
33
23
32
12
21
33
23
31
+
(
)
(3.57)
13
21
32
22
31
Clearly, the determinant of the identity matrix I is 1 for any n since choos-
ing
I I .
We can derive some useful information from studying how elementary row
operations (see Definition 3.3) affect the determinant of a matrix. This provides a
way of evaluating determinants that is usually more efficient than direct applica-
tion of Equations (3.54) and (3.55).
=
1
reduces Equation (3.55) to
det
=
I
det
k
n
11
n
1