Civil Engineering Reference
In-Depth Information
The
adjoint matrix
,
Adj
[
A
], is simply the transpose of the cofactor matrix.
This can be expressed in a few ways.
T
T
=−
+
(
ij
)
AdjA CA c
[][] [
=
=
]
Or
() []
1
A
cofactor
ij
ji
T
ccc
c
ccc
c
11
12
13
1
m
11
21
31
n
1
ccc
c
ccc
c
21
22
23
2
m
12
22
32
n
2
[]
=
AdjA
ccc
c
=
ccc
c
31
32
33
3
m
13
23
33
n
3
ccc
c
ccc
c
n
1
n
2
n
3
nm
1
m
2
m
3
m
mn
It is noted that the subscripts of the adjoint matrix are the reverse of the
cofactor matrix. The main difference is that the transpose is performed
during the operation of taking the adjoint, while in the cofactor method is
done at the end.
Example 2.5
Cofactor method
Find the solution set to the following nonhomogeneous linear algebraic
equations using the cofactor method.
+++=
+++=
−+−+=
+++
x
xxxx
x x xx
xxxx
x
10
1
2
3
4
842
26
1
2
3
4
2
3210
1
2
3
4
x
x
=
10
1
2
3
4
=
[]
T
C
A
=
()
ij
+
−
1
A
And
C
1
A
ij
ij
The determinants are shown by row expansion for the 4 × 4 matrix and by
the basket weave for all the 3 × 3 matrices in Table 2.2.
The last step in Table 2.2 is to multiply the invert of
A
, [
A
]
−1
, times the
constant vector, [
C
], to get the final solution vector, [
x
].
Example 2.6
Method of adjoints
Determine the solution to the following set of equations using the adjoint
method. Use the basket weave method for determinants.
