Game Development Reference
In-Depth Information
Let's briefly state some important characteristics concerning determi-
nants.
•
The determinant of an identity matrix of any dimension is 1:
|
I
| = 1.
Determinant of identity
matrix
•
The determinant of a matrix product is equal to the product of the
determinants:
|
AB
| = |
A
||
B
|.
Determinant of matrix
product
This extends to more than two matrices:
|
M
1
M
2
M
n−1
M
n
| = |
M
1
||
M
2
| |
M
n−1
||
M
n
|.
•
The determinant of the transpose of a matrix is equal to the original
determinant:
M
T
= |
M
|.
Determinant of matrix
transpose
•
If any row or column in a matrix contains all 0s, then the determinant
of that matrix is 0:
?
?
?
Determinant of matrix
with a row/column full
of 0s
= 0.
?
?
?
?
?
0
?
.
.
.
?
?
0
?
=
.
.
.
.
0
0
0
.
.
.
?
?
0
?
?
?
?
•
Exchanging any pair of rows negates the determinant:
m
11
m
12
m
1n
m
11
m
12
m
1n
Swapping rows negates
the determinant
m
21
m
22
m
2n
m
21
m
22
m
2n
.
.
.
.
.
.
m
i1
m
i2
m
in
m
j1
m
j2
m
jn
= −
.
.
.
.
.
.
.
m
j1
m
j2
m
jn
m
i1
m
i2
m
in
.
.
.
.
.
.
m
n1
m
n2
m
nn
m
n1
m
n2
m
nn
This same rule applies for exchanging a pair of columns.
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