. .
1. 1 7 0
2 -1 -3 = 2 - 21*5 + 0 0 - (-3)*41 - 14*2 = 2 -105+12-28 =14-133=-119
5 4 2
2. + + +
4 6 -2 1 (-4) (- 6) (2) 0 0 0 1
1 -1 3 5 = -19 -31 13 5 19 31 13
3 -10 2 1 -1 -16 4 1 = -1 1 16 4 =
6 0 12 2 -2 -12 16 2 2 12 16
19 31 13 19 31 13 19 31 13
= -2 1 16 4 = -2 1 16 4 = -2 1 26 0 =
1 6 8 0 -10 4 0 -10 4
= -2 * (12*26*4 + 130 31*4) = -4 * (12*26*2 + 65 31*2) = -4 * (12*26*2 + 3)
. , .
. , ∆ = |A|=0.
2. .
b a, ab=1 b= = a-1 aa-1 ≡ 1
. , : AB=BA=E |
=-1.
A-1A=AA-1=E
| -1|=| -1|||
| -1|*||= 1, |-1|= , ||≠ 0
|-1|=| |= 1
, , ││≠ 0. . |A|=0, ().
.
││≠ 0 - .
. , -1 , : *-1 = -1 * = |
.
1. |A|. ││≠ 0, .
2. .
11 12 13
= 21 22 23
31 32 33
3.
11 21 31
12 22 32 = Ã
13 23 33
.
4. |A|.
│-1│ = * Ã
. │-1│ = * Ã = ,
. ,
4 -8 -5
= -4 7 -1
-3 5 1
1. 4 -8 -5 24 -43 0
|| = -4 7 -1 = -4 7 -1 = 24*12- 7*43 = 288 301 = -13 -3 5 1 -3 5 1-3 5 1 -7 12 0
-7 12
2. 7 -1 -4 -1
11 = = 7 + 5 = 12, 12 = - = 7, 13 = 1,
5 1 -3 1
-8 -5 4 -5
21 = - = -17, 22 = = -11, 23= 4,
5 1 -3 1
31 = 43, 32= 24 33= - 4
12 7 1
-17 -11 4;
43 24 -4
3. 12 7 1
à = -17 -11 4;
|
|
43 24 -4
4.
- -
-1 = - -
- -
.
- .
m n. , .
11 12 13 14 - 3-
21 22 23 24
31 32 33 34 - 18 2- .
:
11 12 22 23
21 22 32 33
. , 3-
11 12 13
21 22 23 .
31 32 33.
. , , rang A=r. |
: ) rang A≤ min (m, n);
) , rang A=n;
.
1 4 7 12
= 3 -1 0 13 rang ≤ min (3;4)
4 16 28 48
3 (.. 3- 1- ).
2- 1 4 = - 13 ≠ 0
3 -1
2- . , rang =2.
. , . . :
1. , , .
2. .
3. , () .
4. .
5. , , , .
.
.
1 3 7 2 5 + (3) (-2)
-1 0 4 8 3
3 6 10 -4 7
2 -3 10 4 4
1 3 7 2 5 1 3 7 2 5
0 3 11 10 8 => 0 3 11 10 8 (3)
0 -3 -11 -10 -8 0 -9 -4 0 -6
0 -9 -4 0 -6
1 3 7 2 5 1 3 7
0 3 11 10 8 = 0 3 11 = 1*3*29 = 87≠ 0
0 0 29 30 18, 0 0 29
rang A45 = 3.