1.
. .
n × m , , n m .
. | (1.1.1) |
aik (i = 1, 2, , n; k = 1, 2, , m), , . , .
n m , , n ´ m.
, , .
: , .
, , () , () . ,
,
.
n = m, .
, n n , n .
, , , , , .
, , E.
. | (1.1.2) |
A . A *.
, .
, (A *)* = A. , (A * = A), .
1. .
; . | (1.2.1) |
A B C = A + B , A B, ..
. | (1.2.2) |
, , .
2. .
A l C = l A, A, l, ..
, (i = 1, 2, , n; k = 1, 2, , m). | (1.2.3) |
A = (1)× A A. , A + ( A) = 0, 0 , A.
, , , :
(1.2.4) | |
3. .
A B C = A B, B A.
(1.2.5) |
C = A B, A B:
|
|
(i = 1, 2, , n; k = 1, 2, , m). | (1.2.7) |
4. .
A B, , = A × B, cik i A k B.
A B , A B.
; . A × B.
.
BA , , .. (2) B ( ) (3) A ( ).
A × B B × A.
, .
, A × B ¹ B × A, .. .
:
(1.2.8) | |
1.3. 2 3
, .
A 2 :
. | (1.3.1) |
A 2 , :
(1.3.2) |
3 :
(1.3.3) |
A 3 , :
(1.3.4) |
1
, . .:
det A * = det A | (1.4.1) |
2
( ) . .
(1.4.2) |
3
, , :
(1.4.3) |
4
, , :
(1.4.4) |
5
:
(1.4.5) |
6
, , . :
(1.4.6) |
7
, , :
(1.4.7) |
8
, , .
(1.4.8) |
Mik aik , i k , .
Mik.
M 11 a 11=5 :
M 32 a 32=1 :
Aik aik , +, i + k , , i + k , ..
|
|
Aik =(1) i + k Mik. | (1.4.9) |
A 11 = (1)1+1 M 11 = M 11, A 12 = (1)1+2 M 12 = M 12, A 32 = (1)3+2 M 32 = M 32 ..
9
, .. (1.3.3)
det A = a 11 A 11 + a 12 A 12 + a 13 A 13; det A = a 12 A 12 + a 22 A 22 + a 23 A 23 .. | (1.4.10) |
:
det A = a 11 A 11 + a 12 A 12 + a 13 A 13.
, det A = 3×7 + (2) ×(10) + 0×(5) = 41.
10
:
a 11 A 21 + a 12 A 22 + a 13 A 13 = 0, a 11 A 13 + a 21 A 23 + a 31 A 33 = 0 .. | (1.4.11) |
n .
A n n! n , .
, n (n!). , n = 4 24 . .
, 4, . 4, 5 9, . , 4 3 . , 8, .
( )
() :
(1.5.1) |
(1.5.1) i . .
.
A 4 :
9 det A, , , 3 :
: 2 2 3 ; 2 6 4 .
:
(1.6.1) |
B A, A × B = B × A = E, E .
, A, A 1.
, , det A ¹ 0, A A 1, :
(1.6.2) |
Aik aik A (i, k = 1, 2, , n). :
1. A , A .
2. A det A.
3. .
, :
A 1, .
A det A , :
A 11 = 4, A 12 = (2) = 2, A 21 = 1, A 22 = 3. ,
:
A 1 .
(1.1.1), n m . k . k . , k , m n, . . k £ min(n; m). , , . , .
|
|
. rang A r(A).
:
1. A , .. k £ min(n; m).
2. , , .. A =0.
3. n r (A) = n , A .
A 4´3, rang A £ 3. , A . , rang A £ 2. , , , 2.
, , . , , .
A , r (A) £ 4. |A| = 0, A , r (A) £ 3. r (A) £ 2. ( ), ( ), ; r (A) £ 1. A , .. , r (A) = 1.
. , .
1. ().
2. () , .
3. (), () ().
4. .
.
1.8. .
(1.8.1) |
n m x 1, x 2, xm. aij bj (i = 1, 2, , n; j = 1, 2, , m) , (1.8.1).
(1.8.2) |
.
, bi (i = 1, 2, , n) , .
x 1 = a1, x 2 = a2, , x m = am, (1.8.1) .
, , . , , . , , , , .
, . , , .
|
|
:
x 1 = 0, x 2 = 0, , xm = 0. | (1.8.3) |
, .
1.9. n n .
, (n = m):
(1.9.1) |
, ( ) :
(1.9.2) |
, , , :
(1.9.3) |
, | (1.9.4) |
Bi (i = 1, 2, , n) A i .
:
:
B 1, B 2, B 3:
, :
.