Graphics Reference
In-Depth Information
3.1.3
Determinants
The
determinant
of a matrix
A
is a number associated with
A
, denoted det(
A
)or
|
. It is often used in determining the solvability of systems of linear equations,
as discussed in the next section. In this topic the focus is on 2
A
|
×
2 and 3
×
3
determinants. For matrices up to a dimension of 3
×
3, determinants are calculated as
follows:
|
A
| = |
u
1
| =
u
1
,
=
u
1
u
2
|
A
| =
u
1
v
2
−
u
2
v
1
, and
v
1
v
2
u
1
u
2
u
3
v
1
v
2
v
3
w
1
w
2
w
3
|
|
=
A
=
u
1
(
v
2
w
3
−
v
3
w
2
)
+
u
2
(
v
3
w
1
−
v
1
w
3
)
+
u
3
(
v
1
w
2
−
v
2
w
1
)
=
u
·
(
v
×
w
)
.
(The symbols
are the dot product and cross product, as described in Sections
3.3.3 and 3.3.5, respectively.) Determinants are geometrically related to (oriented)
hypervolumes
: a generalized concept of volumes in
n
-dimensional space for an
n
·
and
×
n
determinant, where length, area, and volume are the 1D, 2D, and 3D volume mea-
surements. For a 1
×
×
1 matrix,
A
=
[
u
1
],
|
A
|
corresponds to the signed length of a
line segment from the origin to
u
1
.Fora2
×
2 matrix,
u
1
,
u
2
A
=
v
1
v
2
|
|
corresponds to the signed area of the parallelogram determined by the points
(
u
1
,
u
2
), (
v
1
,
v
2
), and (0, 0), the last point being the origin. If the parallelogram is swept
counterclockwise from the first point to the second, the determinant is positive, else
negative. For a 3
A
×
=
3 matrix,
A
[
uvw
] (where
u
,
v
, and
w
are column vectors),
|
corresponds to the signed volume of the parallelepiped determined by the three
vectors. In general, for an
n
-dimensional matrix the determinant corresponds to the
signed hypervolume of the
n
-dimensional hyper-parallelepiped determined by its
column vectors.
Entire topics have been written about the identities of determinants. Here, only
the following important identities are noted:
A
|
The determinant of a matrix remains unchanged if the matrix is transposed,
●
| =
A
T
.
|
A