A λ (: λ A) B, A , B
1. 1*A = A;
2. (Λβ)A = Λ(βA)
3. (Λ+β)A = ΛA + βA
4. Λ(A+B) = ΛA + ΛB
A + B C, A B, C
5.;
6.;
7. ;
8. ;
:
MxN P( ), .
(: AB, ) C, .
A B. A , B , AB = C .
1.;
2. ;
3. ;
4. ;
5.(ΛA)B = Λ(AB) = A(ΛB);
A = (aij) , ( ! . ) . , a.
: A = (aij), AT = (aji). : . , , , .
: , , .
: n- n×n.
. , .
, .
: , .
|
|
: ( I) .
: , .
(=), (.. ). ,
=B, a11=b11, a12=b12, a21=b21, a22=b22
A n D, . n- :
det[A] = |
| = Σ(-1)e a1α1a2α2... anαn (1) |
a1α1a2α2... anαn aij . , . , , .
, , (α1, α2,..., αn) 1 n. n n! (n ), ; a1α1a2α2... anαn ( ). , (-1)e - (α1, α2,..., αn). (-1)e sgn(α), (α1, α2,..., αn).
. aij ||, (-1)ea1α1a2α2... anαn - .
. 2- :
det[A] = |
| = a11a22 - a12a22 (2) |
3- :
det[A] = |
| = a11a22a33 +a12a23a31 +a13a21a32 -a13a22a31 -a11a23a32 -a12a21a33 (3) |
, (1, 2, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1), (1, 3, 2), (2, 1, 3), 0, 2, 2, 3, 1, 1. n- , "" , .
|
|
3. ́ ́ A−1, A E:
, , . . , .
́ ( ) .
n ( ):
:
AX = B, A , B X :
A − 1 , A:
A − 1 A = E, X = A − 1 B. . ( , ) A. A:
.
, B = 0, : AX = 0 ( ) det A = 0. .