Graphics Reference
In-Depth Information
Cramer's rule gives the solution
d
1
b
1
c
1
a
1
d
1
c
1
a
1
b
1
d
1
d
2
b
2
c
2
a
2
d
2
c
2
a
2
b
2
d
2
d
3
b
3
c
3
a
3
d
3
c
3
a
3
b
3
d
3
x
=
,
y
=
,
z
=
,
where
d
d
d
a
1
b
1
c
1
d
=
a
2
b
2
c
2
.
a
3
b
3
c
3
Solving systems of linear equations using Cramer's rule is not recommended for
systems with more than three or perhaps four equations, in that the amount of work
involved increases drastically. For larger systems, a better solution is to use a Gaussian
elimination algorithm. However, for small systems Cramer's rule works well and is
easy to apply. It also has the benefit of being able to compute the value of just a single
variable. All systems of linear equations encountered in this text are small.
3.1.5
Matrix Inverses for 2
×
2 and 3
×
3 Matrices
Determinants are also involved in the expressions for matrix inverses. The full details
on how to compute matrix inverses is outside the range of topics for this topic. For
purposes here, it is sufficient to note that the inverses for 2
×
2 and 3
×
3 matrices
can be written as
u
22
, and
1
det(
A
)
−
u
12
A
−
1
=
−
u
21
u
11
⎡
⎤
u
22
u
33
−
u
23
u
32
u
13
u
32
−
u
12
u
33
u
12
u
23
−
u
13
u
22
1
det(
A
)
⎣
⎦
.
A
−
1
=
u
23
u
31
−
u
21
u
33
u
11
u
33
−
u
13
u
31
u
13
u
21
−
u
11
u
23
u
21
u
32
−
u
22
u
31
u
12
u
31
−
u
11
u
32
u
11
u
22
−
u
12
u
21
From these expressions, it is clear that if the determinant of a matrix
A
is zero, inv(
A
)
does not exist, as it would result in a division by zero (this property holds for square
matrices of arbitrary size, not just 2
3
matrix
A
can also be expressed in a more geometrical form. Let
A
consist of the three
column vectors
u
,
v
, and
w
:
×
2 and 3
×
3 matrices). The inverse of a 3
×
=
uvw
.
A
The inverse of
A
is then given as
=
abc
T
,
A
−
1
