Civil Engineering Reference
In-Depth Information
Basket weave method yields:
346
123
212
34634
12312
21221
A
=
=
=
()()
+
()()
+
()()
()()
+
()()
+
()()
A
32 2432 61 1
−
4 112 331622
A
=−=
42 41 1
2.4
cRAMER'S RuLE
Two common methods to solve simultaneous equations exist. One is the
elimination of unknowns by elementary row operations and the second
involves the use of determinates. One of the methods involving determi-
nates is known as Cramer's rule. This method was published in 1750 by
Gabriel Cramer (1750). The procedure for Cramer's rule in the solution to
n
linear equations with
n
unknowns is as follows:
A
A
A
A
A
A
A
A
1
2
3
n
x
=
,
x
= =…=
,
x
,
,
x
1
2
3
n
ca a
a
aca
a
1
12
13
1
n
11
1
13
1
n
caa
a
aca
a
2
22
23
2
n
21
2
23
2
n
A
=
ca a
a
,
A
=
aca
a
,
3
32
33
3
n
31
3
33
3
n
1
2
ca a
a
aca
a
n
n
2
n
3
nn
n
1
n
n
3
nn
aac
a
aaa
c
11
12
1
1
n
11
12
13
1
aac
a
aaa
c
21
22
2
2
n
21
22
23
2
A
=
aac
a
,
…
, ,
A
=
aaa
c
3
31
32
3
3
n
n
31
32
33
3
aac
a
aaa
c
n
1
n
2
n
nn
n
1
n
2
n
3
n
As you might see, |
A
1
| is the original coefficient matrix, [
A
], with col-
umn one replaced with the constant column matrix, [
c
]. The solution to
n
simultaneous equations by Cramer's rule requires (
n
−1)*(
n
+1)! multi-
plications. In other words, the solution of ten simultaneous equations by
determinants would require (9)*(11!) =359,251,200 multiplications.
