Civil Engineering Reference
In-Depth Information
of a matrix,
C
ij
, is given in the following equation where |
A
ij
| is the deter-
minant of the
minor as defined in Section 2.2.
=
()
+
ij
C
1
A
ij
ij
Therefore, given the matrix [
A
], the cofactor matrix is shown as follows.
The matrix of cofactors should not be confused with the constant matrix
of the original linear algebraic equation, although they have the same vari-
able, [
C
]:
aaa
a
11
12
13
1
m
aaa
a
21
22
23
2
m
[]
=
A
aaa
a
31
32
33
3
m
a
nm
aaa
n
1
n
2
n
3
CCC
C
11
12
13
1
m
CCC
C
21
22
23
2
m
[]
=
[]
=
CA
CCC
C
31
32
33
3
m
cofactor
CCC
C
n
1
n
2
n
3
nm
Once the cofactor matrix is known, the invert can be easily calculated:
=
[]
T
C
A
−
1
A
The solutions to the simultaneous equations are now found from matrix
multiplication.
−
1
[ [] []
Ax C
=
Or
[
A Cx
][][]
=
Similar to the cofactor method, the method of adjoints,
Adj
[
A
], is another
common way to solve for the invert of a matrix. This method is as follows:
[]
AdjA
A
−
1
=
A
