Civil Engineering Reference
In-Depth Information
2.2.6
INVERTED MATRIX
Only square matrices where the determinant is not equal to zero (|
A
| ≠ 0)
can have an
inverse
or reciprocal. These matrices are called non-singular,
which implies that reciprocals of rectangular matrices do exist. The inverse
of a matrix is defined as follows:
[]
=
[[
−1
I AA
2.2.7
MATRIX MINOR
The
matrix minor
, [
A
ij
], is found by omitting the ith row and the jth column
of a matrix and writing the remaining terms in a matrix of one size smaller
in rows and columns. It is used in the computation of the determinant. For
example, the minor, [
A
22
] is shown in the following. Note that
i
= 2 and
j
= 2:
a
a
a
a
11
12
13
1
m
aa a
aa a
11
13
1
m
aaa
a
21
22
23
2
m
[
]
=
31
33
3
m
A
a
a
a
a
=
31
32
33
3
m
22
aa a
n
1
n
3
nm
a
a
a
a
n
1
n
2
n
3
nm
2.2.8
TRANSPOSED MATRIX
The
transposed matrix
is found by writing the
a
ij
elements of a matrix as
the
a
ji
elements of the matrix, [
A
]
T
.
aaa
a
11
12
13
1
m
aaa
a
21
22
23
2
m
[]
=
A
aaa
a
31
32
33
3
m
a
aaa
n
1
n
2
n
3
nm
aaa
a
11
21
31
n
1
aaa
a
12
22
32
n
2
[]
=
T
aaa
a
A
13
23
33
n
3
aaa
a
1
m
2
m
3
m
mn
