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):

ab
ad

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

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