Gauss method

1. Gauss method

GAUSS METHOD

2.

a11 x1 a12 x2 ... a1n xn b1
a x a x ... a x b
21 1 22 2
2n n
2
a11
am1 x1 am 2 x2 ... amn xn bm ~ a21
A
a
b1
a
a
am1
1n
11 12
0 a ' a ' b 2'
22
2n
'
0
'
0 amn
b
m
a12
a22
am 2
a1n b1
a2 n b2
amn bm

3.

1 ≠0 & b2 ≠0 => nonhomogeneous system
5 x 3 y 2 z 1 b∆=-716≠0=>
nondegenerate matrix & consistent
system (has solution ) & definite system (has the only
x
10
y
5
z
2
solution )
9 x y 8 z 0
5
3
2 1
~
A 1 10 5 2
9 1 8 0
1 10 5 2
I II
5
3
2 1
9 1 8 0

4.

1 10 5 2
1 10
5 2
I II
I 5 II II
5
3
2 1
0 53 23 11
I ( 9 ) III III
9 1 8 0
0 91 37 18
1 10 5 2
II 91 III ( 53) III
0 53 23 11
0
132
z
47
0
132
47
x 10 y 5 z 2
47
7
x 2 5
10
132
132
41
x
132
53 y 23z 11
47
132
y
53
47
11 23
132
y
53
7
y
132
7 47
41
Ответ :
;
;
132
132
132
11 23
z
47
132
b1 ≠0 & b2 ≠0 =>
nonhomogeneous system
∆=-716≠0=>
nondegenerate matrix &
consistent system (has
solution ) &
definite
system (has the only
solution )

5.

9a 2b 7c d 8
7a 9b 7c 4d 3
9a 4b 9c 5d 1
15a 2c 2d 6
b1 ≠0 & b2 ≠0 =>
nonhomogeneous system
∆=0=> degenerate matrix
9 2
7
1 8
I ( 7 ) II 9 0
95 112 43 83
I III
0
2
16
6
9
I 15 IV 9
0 30 87
33
66
9
2 7
7 9 7
9
4
9
15 0 2
18
43
51
2 6
9 2
7
1 8
II 2 III 95 0
95 112 43 83
II 30 IV 95 0
0 1744 656 1021
0 0
4905 1845 3780
9 2
7
1 8
III 4905 IV 1744 0
95 112 43 83
0 0 1744 656 1021
0 0
0
0
1584315
0 d 1584315
inconsistent system (has no
solution )

6.

4 x1 9 x2 9 x3 5 x4 6
9 x 9 x 2 x 9 x 0
1
2
3
4
3 x1 9 x2 6 x3 3
27 x1 3 x2 4 x3 x4 2
b1 ≠0 & b2 ≠0 =>
nonhomogeneous system
∆=0=> degenerate matrix
4
9
9
5 6
9 2 9 0
9
3 9 6 0 3
27 3 4 1 2
4 9
9
5 6
I 9 II 4 0
45 89 81 54
I 3 II 9 III 4 0
63
51
15
30
I 27 IV 4
0 255 259 131 154
4 9
9
5 6
II 63 III 45 0
45
89
81 54
II 255 IV 45 0
0 3312 4428 2052
0
0 11040 14760 6840
9
5 6
4 9
consistent system (has solution )
III 11040 IV 3312 0
45 89
81 54 & indefinite system (has more
0
0 3312 4428 2052 than one solution )
0
0
0
0 0
0 x4 0

7.

4 9
9
5 6
81 54
0 45 89
0
0 3312 4428 2052
0
0
0
0 0
0 x4 0
x4 t
3312 x3 4428t 2052
57 123t
x3
92
57 123t
45 x2 89
81t 54
92
7 233t
x2
276
7 233t
57 123t
4 x1 9
9
5t 6
276
92
15 13t
x1
92
15 13t 7 233t 57 123t
;
;
; t general solution
92
276
92
7
57
15
;
;
;
0
92
92 276
60
45 particular solutions
7
;
;
;1
23
23 69

8.

3a b c d 3
a 8b 3c 6
7 a b c d 8
3 1 1
1 3
1 8 3 0 6
7 1 1 1 8
1 8 3 0 6
1 8 3 0 6
I 3 II II
I II
0 23 10 1 15
3 1 1
1 3
I 7 III III
0 55 20 1 34
7 1 1 1 8
1 8 3
0 6
d t
II 55 III 23 III
90c 78d 43
0 23 10 1 15
0 0 90 78 43
43 78t
c
90
23b 10c d 15
92 69t
43 78t
43 78t
a 8
3
6
23b 10
t 15
207
90
90
2093 138t
92 69t
a
b
2070
207
2093 138t 92 69t 43 78t
2369 46 199
;
;
;t
;
;
;2
2070
207
90
2070 207 90