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a11 x1 + a12 x2 + …+ a1n xn = 0 a21 x1 + a22 x2 + …+ a2n xn = 0 …
…
…
…
am1 x1 + am2 x2 + …+ amn xn = 0
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E1: 11x1 - 3x2 E2: E3:
-3x1 + 6x2
= 30 - x3 =
5
-x2 + 3x3 = -25
3E1: 11E2:
33x1 - 9x2
-33x1 + 66x2 -11x3 = 55
E’2:
57x2 -11x3
E1: 11x1 - 3x2
57x2 -11x3 = 145
E3:
-x2 + 3x3 = -25
E’2:
57x2 -113x3 = 145
E3:
-57x2 +171 x3 = -25*57
.
160 x3 = -25*57+145
E1: 11x1
-3x2
E’2:
57x2 -11x3 = 145
E’3:
= 145
= 30
E’2:
E’3:
= 90
= 30
160x3 = -25*57+145
∙
E1: x1
- x3 = 30
E2: 3x1 + 5x2 + 2x3 =
5
E3: 4x1 + 7x2 + 3x3 = -25
-3E1: -3x1
+ 3x3 = -90
E2: 3x1 + 5x2 + 2x3 = E’2:
5
5x2 + 5x3 = -85
-4E1: -4x1 + 4x3 = -120 E1: x1 - x3 = 30 E3: 4x1 + 7x2 + 3x3 = -25 E’2: 5x2 + 5x3 = -85 E’3: 7x2 + 7x3 = -145 E’3: 7x2 + 7x3 = -145
E1: x1
- x3 =
30
E’’2:
x2 + x3 = -85
E’’3:
x2 + x3 = -145
E1: x1
- x3 = 30
E2: 3x1 + 5x2 + 2x3 = 5 E3: 4x1 + 7x2 + 3x3 = 1
E1: x1
- x3 =
30
E’2:
5x2 + 5x3 = -85
E’3:
7x2 + 7x3 = -119
E1: x1
- x3 = 30
E’’2:
x2 + x3 = -17
E’’3:
x2 + x3 = -17
x1 = x3 + 30 x2 = -x3 - 17
a11 x1 + a12 x2 + …+ a1n xn = b1 a21 x1 + a22 x2 + …+ a2n xn = b2 …….……………………………... am1 x1 + am2 x2 + …+ amn xn = bm
a11 x1 +
a12 x2 + ……………. + a1n xn = b1
(a21 + r a11)x1 + (a22 + r a12) x2 + … + (a2n + r a1n) xn = b2 + rb1 …….……………………………………………………………….... am1 x1 +
am2 x2 + ……………. + amn xn = bm
a11 x1 + a12 x2 + … + a1n xn = b1 a22’ x2 + … + a2n’ xn = b2’ …..……………………..... am2’ x2 + … + amn’ xn = bm’
x1 + a12” x2 + a13” x3 + ….. + a1n” xn = b1” x2 + a23” x3 + ….. + a2n” xn = b2” …..……………………............. x2 + am3” x3 + … + amn” xn = bm”
1x1 + 0x2 + d13 x3 + ….. + d1n xn = c1 0x1 + 1x2 + d23 x3 + ….. + d2n xn = c2 ………..…..…………………….......... 0x1 + 0x2 + dm3 x3 + … + dmn xn = cm
1x1 + 0x2 + 0x3 + …. + 0xm + d’1,m+1 xm+1 + … + d’1,m+t xm+t = c’1 0x1 + 1x2 + 0x3 + ….+ 0xm + d’2,m+1 xm+1 + … + d’2,m+t xm+t = c’2 0x1 + 0x2 + 1x3 + … + 0xm + d’3,m+1 xm+1 + … + d’3,m+t xm+t = c’3 …………………………………..………..……………………. 0x1 + 0x2 + 0x3 + … + 1xm + d’m,m+1 xm+1 + … + d’m,m+txm+t = c’m
x1 = c’1 - (d’1,m+1 xm+1 + … + d’1,m+t xm+t) x2 = c’2 - (d’2,m+1 xm+1 + … + d’2,m+t xm+t) ..………..……………………………..…. xm = c’m - (d’m,m+1 xm+1 + … + d’m,m+txm+t)
1x1 + 0x2 + 0x3 + …. + 0xr + d’1,r+1 xr+1 + … + d’1,n xn = c’1 0x1 + 1x2 + 0x3 + ….+ 0xr + d’2,r+1 xr+1 + … + d’2,n xn = c’2 0x1 + 0x2 + 1x3 + … + 0xr + d’3,r+1 xr+1 + … + d’3,n xn = c’3 …………………………………..………..………………….. 0x1 + 0x2 + 0x3 + … + 1xr + d’r,r+1 xr+1 + … + d’r,nxn = c’r
n
x1 = c’1 - (d’1,r+1 xr+1 + … + d’1,n xn) = c’1 -
d’1,jxj
j r 1
n
x2 = c’2 - (d’2,r+1 xr+1 + … + d’2,n xn) = c’2 -
d’2,jxj
j r 1
..………..……………………....…………………… n
xr = c’r - (d’r,r+1 xr+1 + … + d’r,n xn) = c’r -
j r 1
d’r,jxj
n
j r 1
n
n
j r 1
j r 1
n
n
j r 1
j r 1
xr+1, xr+2 …, xn 1, 0, … , 0 0, 1, … , 0 0, 0, … , 1 x1 = -d1,r+1 x2 = -d’2,r+2 …………… xr = -d’r,n
n
j r 1
1, 2, … , r, r+1, … , n Ar+1 = (-d’1,r+1, -d’2,r+1, … , -d’r,r+1, 1, 0, …, 0) Ar+2 = (-d’1,r+2, -d’2,r+2, … , -d’r,r+2, 0, 1, …, 0) …………………………………………………. An = (-d’1,n, -d’2,n, … , -d’r,n, 0, 0, …, 1)
n
∙
∙
∙
j r 1
ε a11 x1 + a12 x2 + …+ a1n xn = 0 a21 x1 + a22 x2 + …+ a2n xn = 0 …….……………………………... am1 x1 + am2 x2 + …+ amn xn = 0
n
j r 1
n
j r 1
n
j r 1
n
j r 1
n
j r 1
∙
∙
a11 x1 + a12 x2 + …+ a1n xn = b1 a21 x1 + a22 x2 + …+ a2n xn = b2 …….……………………………... am1 x1 + am2 x2 + …+ amn xn = bm
a11 y1 + a12 y2 + …+ a1n yn = 0 a21 y1 + a22 y2 + …+ a2n yn = 0 …….……………………………... am1 y1 + am2 y2 + …+ amn yn = 0
∙
1x1 + 0x2 + 0x3 + …. + 0xm = c’1 0x1 + 1x2 + 0x3 + ….+ 0xm = c’2 0x1 + 0x2 + 1x3 + … + 0xm = c’3 ………………………………….. 0x1 + 0x2 + 0x3 + … +1xm = c’m