Gauss Jordan Method

1. The Gauss-Jordan method
Algorithm
The Gauss-Jordan method transforms the initial expanded matrix into a identity
matrix and a right side equal to the solution that is sought:
↔
nn
cx
.....
cx
cx
=
=
=
22
11
()
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛
=
nnnnn
n
n
b
...
b
b
a...aa
.....
a...aa
a...aa
b|A
2
1
21
22221
11211
↔
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛
n
c
...
c
c
...
.....
...
...
2
1
100
010
001


Example 1. Solve the system using the Gauss-Jordan method with a chosen pivot
element from a row:

7
03253
10524
932
432
4321
321
421
−=−+−
=−++
−=+−
=+−
xxx
xxxx
xxx
xxx
.

Solution: We work the same way as with the Gauss method by choosing a pivot element
from a row but the unknowns are excluded under the main diagonal as well as above it.
The aim is to be left with non-zero elements only along the main diagonal.


)4(:
↔


↔
⎟
⎟
⎟
⎟
⎟
⎠
⎞
−
−
⎜
⎜
⎜
⎜
⎜
⎝
⎛
−−
−
−
−
7
0
9
10
1110
3253
3012
052
⎟
⎟
⎟
⎟
⎟
⎠
⎞
−
−
4
⎜
⎜
⎜
⎜
⎜
⎝
⎛
0
3
−−
−
−
−
7
0
10
9
111
325
052
3012
4



. (-2) .(-3)
↔
⎟
⎟
⎟
⎟
⎟
⎠
⎞
−
−
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
−−
−−
−
−
7
14
1110
30
300
01
2
15
2
5
4
7
2
5
4
5
2
1




↔
⎟
⎟
⎟
⎟
⎟
⎠
⎞
−
−
⎜
⎜
⎜
⎜
⎜
⎝
⎛
−
− 7
0
9
11
32
30
0
2
5
4
51
−
−
−10
53
12
1
2




: )(
2
13
↔
⎟
⎟
⎟
⎟
⎟
⎠
⎞
−
−
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
−−
−−
−−
−
7
14
1110
300
10
01
13
15
2
5
2
5
13
6
26
7
4
5
2
1



.)( .(1)
⎟
⎟
⎟
⎟
⎟
⎠
⎞
−
−
⎜
⎜
⎜
⎜
−0
00
⎜
⎜
⎝
⎛
−
−−
−−
−
7
14
111
3
30
01
2
15
2
5
2
5
4
7
4
5
2
1
4
4
2
13
2
1

↔

13/2

2. ⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
−
−
⎜
⎜
⎜
⎜
⎜
⎝
⎛
−
−−
−
13
76
13
15
13
25
13
19
26
19
13
6
26
7
13
3
26
29
14
00
300
10
01




:)(
2
5
−

↔

-5/2

⎟
⎟
⎟
⎟
⎟
⎠
⎞
−
−
−
⎜
⎜
⎜
⎜
⎜
⎝
⎛
−
−
−−
−
13
76
5
28
13
15
13
25
13
19
26
19
5
6
13
6
26
7
13
3
26
29
00
100
10
01




.)(
26
7
.)(
26
29
−
.)(
26
19
−

↔

⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
−
−
−
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
−
−
−
65
114
5
28
65
23
65
281
65
38
5
6
65
51
65
72
000
100
010
001






:)(
65
38
−

↔

⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
−
−
⎜
⎜
⎜
⎜
⎜
⎝
⎛
−
−
31000
100
010
001
5
28
65
23
65
281
5
6
65
51
65
72





.)(
5
6
.)(
65
51
.)(
65
72
−
↔
⎟
⎟
⎟
⎟
⎟
⎠
⎞
−
⎜
⎜
⎜
⎜
⎜
⎝
⎛
3
2
2
1
1000
0100
0010
0001
↔
3
2
2
1
4
3
2
1=
−=
=
=x
x
x
x



By Iliya Makrelov, [email protected]