Game Development Reference
In-Depth Information
n
()
M ,
det
M
=
MC
kj
kj
j
=
1
()()
ij
+
ij
,
where
C
()
M is the cofactor of i M defined by
C
M
=−
1 t
M
{
}
.
ij
ij
The determinant of a 22
×
matrix is given by
cd =−
bc
,
and the determinant of a 3
×
3
matrix is given by
aaa
aaa aaaaa aaaaa
aaa
11
12
13
(
)
(
)
=
21
22
23
11
22
33
23
32
12
21
33
23
31
(
)
+
aaaaa
31 .
31
32
33
13
21
32
22
Matrix Inverses
An nn
matrix M is invertible if and only if the columns of M form a linearly
independent set. Equivalently, M is invertible if and only if det
×
M
0
.
The entries of the inverse G of an nn
matrix F can be calculated by using the
×
explicit formula
()
det
C
F
ji
G
=
.
ij
F
Using this formula, the inverse of a 22
×
matrix A is given by
A
1
det
A
22
12
1
A
=
,
A
A
A
21
11
and the inverse of a 3
×
3
matrix B is given by
BBBBBBBBBBB
B
22
33
23
32
13
32
12
33
12
23
13
22
1
det
1
B
=
B
BBB BBBB BBBB
.
23
31
21
33
11
33
13
31
13
21
11
23
B
B
BBBBBBBBBBB
21
32
22
31
12
31
11
32
11
22
12
21
Eigenvalues and Eigenvectors
The eigenvalues of an nn
matrix M are equal to the roots of the characteristic
×
polynomial given by