a1,..., an x1,..., xn
x1a1 +... + xnan.
, b a1,..., an, x1,..., xn, a1,..., an b,
x1a1 +... + xnan = b,
x1,..., xn, b a1,..., an.
6 , , . , .
1. : m n -
2. : m=n
3. : m=1. , (1 3 5 7) -
4. : n=1.
5. : m=n aij=0, i≠j.
6. : m=n
7. : aij=0, i=1,2,...,m
8. : 0.
9. : m=n aij=aji (.. ), A'=A
10. : m=n aij=-aji (.. ). , (.. i=j aii=-aii)
11. : m=n aii=-ãii (ãji - - aji, .. A=3+2i, - Ã=3-2i)
A = || ai j || B = || ai j || , :
(1) |
i j.
().
A λ ( ) :
(2) |
. A = || ai j || B = || bi j || C = || ci j ||, :
(3) |
A B , .
7 A n = 1,2,3,...
.
.
A
A - n. n=1, A A. , A, .. .
|
|
n=2, A
,
n=3, A
,
1. .
2. , .
3. , .
4. , , .
5. k, k.
6. , , .
7. i- a i j = b j + c j (j=), , , i-, - , , i- b j, - c j.
8. , , ..
8 ( ).
( ) . , d i-
d = a i 1 A i 1 + a i 2 A i 2 +... + a i n A i n (i =)
j-
d = a 1 j A 1 j + a 2 j A 2 j +... + a n j A n j (j =).
, ( ), , , , .
9 ́ ́ A−1, , A E:
. , A , , , .. det A ≠ 0.
10 ́ ́ A−1, , A E:
: A E. A . . , A−1.
( , ):
11 , .