(). , , . . : , .
. - , , . : , , , . ( ) , , . .
, , .
:
a 11 x 1 | + | a 12 x 2 | + | ... | + | a 1 n xn | = | a 1, n +1 | |
a 21 x 1 | + | a 22 x 2 | + | ... | + | a 2 n xn | = | a 2, n +1 | (3.1) |
.... | . | .... | . | ... | . | .... | . | .... | |
an 1 x 1 | + | an 2 x 2 | + | ... | + | ann xn | = | an,n+1 |
- (i =1,..., n, j =1,..., n +1);
xi - () ;
n - .
(3.1) n n +1:
a 11 | a 12 | ... | a 1 n | a 1, n +1 |
a 21 | a 22 | ... | a 2 n | a 2, n +1 |
. | . | . | . | . |
an 1 | an 2 | ... | ann | an,n +1 |
n , (3.1).
- xi, (3.1) .
xi (3.1) xi = Di / D, D - , Di - , i - .
2-, 3- , , , , .
. , , , . ; .
|
|
3-
a 11 x 1 | + | a 12 x 2 | + | a 13 x 3 | = | a 14 | |
a 21 x 1 | + | a 22 x 2 | + | a 23 x 3 | = | a 24 | (3.1′) |
a 31 x 1 | + | a 32 x 2 | + | a 33 x 3 | = | a 34 | . |
(3.1′) x 1:
x 1 = (a 14 - a 12 x 2 - a 13 x 3) / a 11 , | (3.2) |
:
(3.3) |
, j = 2,3,4. | (3.4) |
(3.2) (3.4) (3.1′) :
(3.5) |
, i =2,3; j = 2,3,4,
.. x 1.
(3.5) x 2:
, | (3.6) |
:
, | (3.7) |
, j = 3,4. | (3.8) |
(3.6) (3.8) (3.5) :
(3.9) |
, j = 3,4,
.. x 2.
(3.9) x 3: ,
.
:
(3.10) |
. :
(3.11) |
. , .
. (3.10) x 3
. | (3.12) |
(3.10), :
. | (3.13) |
x 2, x 3 (3.10), x 1:
. | (3.14) |
(3.12), (3.13), (3.14) (3.1′).
n- . k- (k=1,2,3,...,n) :
, | j = 1,2,..., n +1, | (3.15) |
, | i = k +1,..., n; j = 1,2,..., n +1. | (3.16) |
, .. k=n, (3.15), (3.16) i.
(3.15) akk. , . , k - n - , a kk . , , , akk , k - . . - , , D . , , .. , .
(3.15), (3.16) j = 1,..., n +1, .. x 1,..., x n, .
|
|
, j = 1,..., n, n.
(3.15) (3.16), :
, | (3.17) |
k = 1,..., n -1 - ; i = k +1,..., n; j = 1,2,..., n.
i k +1, , .
(3.17) , , , :
(3.17) ; - , . .
, , , k - kk , k - k - . , , .
, (3.17) :
,
p , .
X A, E:
A . X = E. | (3.18) |
, .
, A X n C , :
(3.19) |
, .. X, (3.18), :
; | ; |
(3.18) (3.19) :
:
a 11 x 11 + a 12 x 21 + a 13 x 31 | = | ||
1) | a 21 x 11 + a 22 x 21 + a 23 x 31 | = | |
a 31 x 11 + a 32 x 21 + a 33 x 31 | = |
a 11 x 12 + a 12 x 22 + a 13 x 32 | = | ||
2) | a 21 x 12 + a 22 x 22 + a 23 x 32 | = | |
a 31 x 12 + a 32 x 22 + a 33 x 32 | = |
a 11 x 13 + a 12 x 23 + a 13 x 33 | = | ||
3) | a 21 x 13 + a 22 x 23 + a 23 x 33 | = | |
a 31 x 13 + a 32 x 23 + a 33 x 33 | = |
, , .
, X, n, n , , - E.
n . n 2 n; , - E.