Civil Engineering Reference
In-Depth Information
Section 2.3. In comparison to a non-trivial solution, a trivial solution is
one where all the unknowns in the equations are equal to zero.
The typical set of
n
equations with
n
unknowns is as follows:
+
…
+
ax ax ax ax C
ax ax ax ax C
a
+
+
=
11 1
122 13
3
1
nn
1
+
…
+
+
+
=
21 1
222 23
3
2
nn
2
+
…
+
xaxax
+
+
a xC
=
3
111 32
2
333
3
nn
3
+
…
+
ax
+
a
n
xax
+
a xC
=
n
11
22 33
n
nn
n
n
These equations can be written in matrix form, [
A
][
x
]=[
C
] as follows:
aaa
a
x
x
x
C
C
C
11
12
13
1
n
1
1
aaa
a
21
22
23
2
n
2
2
aaa
a
=
31
32
33
3
n
3
3
aaa
a
x
C
n
1
n
2
n
3
nn
n
n
2.2 MAtRicES
A
matrix
can be defined as a rectangular array of symbols or numerical
quantities arranged in rows and columns. This array is enclosed in brack-
ets and if there are
n
rows and
m
columns, the general form of this matrix
is expressed by the following:
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
A matrix consisting of
n
rows and
m
columns is defined as a matrix of
order
n
×
m.
The relationship between the number of rows and the number
of columns is arbitrary in a general matrix. Many types of matrices exist,
such as row, column, diagonal, square, triangular, identity, and invert.
These are discussed in the following sub-sections.
