Civil Engineering Reference
In-Depth Information
100
0
aaa a
a
−
1
−
1
−
1
−
1
11
12
13
1
n
010
0
−
1
−
1
−
1
−
1
aa a
aaa
2
1
22
23
2
n
−
1
=
IA
|
001
0
−
1
−
1
−
1
−
1
a
31
32
33
3
n
000 1
−
1
−
1
−
1
−
1
aaa
a
nn
n
1
n
2
n
3
If partial pivoting is used during the reduction, the columns in the
invert must be swapped back in reverse order as the rows were swapped
during reduction. The constant vector must also be reordered in the
same way.
Example 2.13
Matrix inversion
Determine the solution to the following set of equations using inversion-
in-place (improved Gauss-Jordan). Include partial pivoting during the
reduction.
211211
.
x
+
.
x
−
304111
.
x
+
.
x
=
165
.
1
2
3
4
−
002123
.
x
+
.
x
+
222102
.
x
+
.
x
=
13 18
.
1
2
3
4
014006
.
x
−
.
x
+
121108
.
x
−
.
x
= −
067
.
1
2
3
4
132020
.
x
+
.
x
+
0
.00
x
+
390 732
.
x
=
.
1
2
3
4
The matrix is shown in Table 2.14 in augmented form and Gauss-Jordan
elimination is performed for the first two columns.
Table 2.14.
Example 2.13 Matrix inversion method
2.11
2.11
1.11
1.65
−3.04
A|C
=
−0.02
1.23
2.22
1.02
13.18
0.14
1.21
−0.06
−1.08
−0.67
1.32
0.20
0.00
3.90
17.32
2.11
2.11
−3.04
1.11
1.00
0.00
0.00
0.00
A|I
=
−0.02
1.23
2.22
1.02
0.00
1.00
0.00
0.00
0.14
1.21
0.00
0.00
1.00
0.00
−0.06
−1.08
1.32
0.20
0.00
3.90
0.00
0.00
0.00 1.00
(
Continued
)
