a 11 x 1 +... + a 1n x n = b 1,
a 21 x 1 +... + a 2n x n = b 2,
........................ ()
a m1 x 1 +... + a mn x n = b m,
aij , bi - , xj - . x 1,..., x n. r 1 ,..., r n (), . , . . , . , .
A = , x = , b =
()
A x = b. ()
,
B =
.
a 1 = ,..., a n =
()
x 1 a 1 +... + x n a n = b.
, .
.
3 , . , () - 1.
x = A- 1 b. ()
, () (). , (), () , () , . , , .
() , xi i- - 1 b. , () 3,
xi = (ï A ç i1b1 +... + ï A ê nibn)
, ,
xi = (i = 1,..., n) ()
() , .
.
x1 - x2 + 2x3 = 7
2x1 + x2 + 3x3 = 4
4x1 + 2x2 + x3 = 3
D = , D 1 = , D 2 = , D 3 =
D, 2- 3- , 2 4.
D = = 1´ = - 21 + 6 = 15.
, D 1 = 30, D 2 = -45, D 3 = 15. ,
x1 = = 2, x2 = = -3, x3 = = 1.
, () . . , , .
|
|
,
a11x1 + a12 x2 +... + a1n xn = b1
a21x1 + a22 x2 +... + a2n xn = b2
.......................... ()
an1 x1 + an2 x2 +... + ann xn = bn
, 11 x1 , . 11
x1 + a12 x2 +... + a1n xn = b1
a21x1 + a22 x2 +... + a2n xn = b2
.......................... ()
an1 x1 + an2 x2 +... + ann xn = bn
2-, 3-,..., n- 1- , a21, a31,..., an1.
x1 + a12 x2 +... + a1n xn = b1
a22 x2 +... + a2n xn = b2
.................... ()
an2 x2 +... + ann xn = bn.
, . , . ,
x1 + w 12 x2 +... + w 1n xn = d1
x2 +... + w 2n xn = d2
................ ()
x n-1 + w nn xn = d n-1
xn = d n
. x n. x n-1 . . x1,..., x n.
.
x1 + 3x2 + x3 = 14
2x1 + 8x2 - x3 = 22
x1 + 6x2 + 2x3 = 27
2- 3- 1- , 2 1.
x1 + 3x2 + x3 = 14
2x2 - 3x3 = - 6
3x2 + x3 = 13.
x1 2- 3- . x3 3- 2- 2
x1 + 3x2 + x3 = 14
x2 - 1.5x3 = -3
3x2 + x3 = 13.
3- 2-, 3.
x1 + 3x2 + x3 = 14
x2 - 1.5x3 = -3
5.5x3 = 22.
3- 5.5, x3 = 4. x3 2- , x2 - 1.5´ 4 = -3 x2 = 3. 1- , x1 + 3´3 + 4 = 14 x1 = 1.
. , xk . . .
.
x1 + x2 - 2x3 = 4
3x1 + x2 + x3 = 9
2x1 + x2 - x3 = 6
.
. . 2- 3- , 3 2.
|
|
.
3- -1 2- .
.
1- 3- , 1 -2.
.
, 1- 2- , 1 -3.
.
x1 = 1, x2 = 5, x3 = 1.