.
, . :
11 121n
21 222n , , = (ij) m,n
= ..
m1 m2mn
ij , i- , j- ;
m ;
n .
m, n (m , n ).
m=n, n- .
, , , , , n, n . ,
1 0 0
3 = 0 1 0 - 3- .
0 0 1
=+, . . ,
11 12 11 12 11+11 12+12
= 21 22 , = 21 22, = + = 21+21 22+22 .
31 32 31 32 31+31 32+32
, . . ,
11 12 11 12
= 21 22, = = 21 22.
31 32 31 32
. , - -.
11
= × = 11 12 13 × 21.
31
= 11 , 1- 2- :
11 = 1111 + 1221 + 1331.
.
11 12 13 11 12 13
=, = 21 22 23.
21 22 23 31 32 33
, :
× = .
m ×p p× nm × n
, , . , , . :
11 12 13 11 12 13 11 12 13
× 21 22 23 =.
21 22 23 31 32 33 21 22 23
2 ×3 3× 32 × 3
11 1- 1- :
11 = 1111 + 1221 +1331.
12 1- 2- :
12 = 1112 + 1222 +1332.
13 1- 3- . , 1- , 2- 3- .
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1-, 2- 3- , :
11 12 13 11 12 13 11 12 13
× 21 22 23 =.
21 22 23 31 32 33 21 22 23
21 = 2111 + 2221 +2331.
, :
× ≠ ×
i- i- . .
: : (C 2E3)2 + RQT.
( C, Q, R 1).
: (C 2E3)2 + RQT =
2
3 2 0 1 0 0 7 0 6 3
= -4 5 1 - 2× 0 1 0 + 2 -4 × 2 0 =
-2 3 4 0 0 1 1 3 1 -4
1 2 0 1 2 0 7 0 6 2 1
= -4 3 1 × -4 3 1 + 2 -4 × 3 0 -4 =
-2 3 2 -2 3 2 1 3
-7 8 2 42 14 7 35 22 9
= -18 4 5 + 0 4 18 = -18 8 23
-18 11 7 15 2 -11 -3 13 -4.
.
() .
11 12
=
21 22
, 1122 2112.
:
1112
det A, ,, 2122 , .
, , , , , , , , , , . ,
2 -1
0 3
2- , . 2, -1, 0, 3. 2×3 0×(-1) = 6.
- , , .
ij Mij.
11 12 13
, 21 22 23
31 32 33
11 12 13 22 23 21 23
11 = 21 22 23 =, 12 = ..
31 32 33 32 33 31 33
ij ij = (-1)i+j Mij.
, :
22 23
11 = (-1)1+1 11 = 11 =,
32 33
21 23
12 = (-1)1+2 12 = - 12 = , ..
31 33
, :
- () .
:
2 4 6 3 0 1 0 1 3
1 3 0 = 2×(-1)1+1 + 4×(-1)1+2 + 6×(-1)1+3 =
-5 2 0 2 0 -5 0 -5 2
2×0 4×0 + 6×17=102.
. , :
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2 4 6 1 3
1 3 0 = 6×(-1)1+3 + 0 + 0 = 102.
-5 2 0 -5 2
, . , , .
( ). .
:
1. det A = det AT.
2. , , ( ), , :
11 12 21 22
21 22 = - 11 12.
3. - () k k, :
11 12 11 12
k21 k22 = k 21 22.
4. :
) - ()
) - () , .
5. , - , , :
11 12 13 11 12 13
21 22 23 = 21 22 23.
31 32 33 31+k11 32+ka12 33+ka13
.
, .
1 1 1 1 2 3
1: 2 2 2, 1 2 3 = 6.
3 3 3 1 2 3
: , .
1 2 3 1 1 1
= 1 2 3, 2 2 2 = det AT = det A = 6.
1 2 3 3 3 3
2: 64 5
32 15
: 1- 32, 2- 5:
64 5 2 1
= 32×5 = 160×5 = 800
32 15 1 3
2. .
, ( ):
111 + 122 ++ 1nn = 1
211 + 222 ++ 2nn = 2 (1)
n11 + n22 ++ mnn = n
, (1) :
11 + 12 ++ 1n 1 1
21 + 22 ++ 2n × 2 = 2 (2)
n1 + n2 ++ mn n n
A×X = B, (3)
A ;
;
.
:
11 + 12 ++ 1n
21 + 22 ++ 2n (4)
=
n1 + n2 ++ mn
:
- , .
- , (), .
:
- ;
- ;
- .
≠ 0, (1) . = 0, , .
≠ 0, (1) , :
1= 1 , 2= 2, ,n= n.
1 (4) 1- ; 2 , 3 . .:
112 1n 111 1n 1112 1
222 2n 212 2n 2122 2
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1= .., 2= .., n =
nn2 mn n1n mn n1n2 n
, , , 1, 2, 3, , , z, .
x + 2y + 3z = 14
: 2x + y =13.
2x + 3y 4z = 23
.
1 2 3 14 2 3 1 14 3 1 2 14
= 2 1 0 =24; = 13 1 0 =96; = 2 13 0 =120; Z= 2 1 13 =0.
2 3 -4 23 3 -4 2 23 -4 2 3 23
= 24 ≠ 0, :
= = 96/24 =4; = y = 120/24 = 5; z = z = 0/24 = 0.
, = 4; = 5; z = 0.
.
, . , , . , , .
-, , , , , , , , , .
:
1 ;
2 , ;
3 , ;
4 ( 0 = 0).
( -) .
3 + 4z = 5
: 2 + z = 5.
+ 2 + 2z = 4
. :
3 -1 4 5
2 1 -1 5
1 2 2 4
, , , . , , . , :
3 -1 4 5 1 2 2 4 1 2 2 4
2 1 -1 5 (1) 2 1 -1 5 (2) 0 -3 -5 -3 (3)
1 2 2 4 3 -1 4 5 0 -7 -2 -7
1 2 2 4 1 2 2 4 1 2 2 4
0 3 5 3 (4) 0 3 5 3 (5) 0 1 -8 1 (6)
0 7 2 7 0 1 -8 1 0 3 5 3
1 0 18 2 1 0 18 2 1 0 0 2
0 1 -8 1 (7) 0 1 -8 1 (8) 0 1 0 1
0 0 29 0 0 0 1 0 0 0 1 0
(1) 1- 3- (.. 1- 3- );
(2) 1- . 2- 1-, -2, 3- 1-, -3 (.. 2- 1-, -2, 3- 1-, -3);
(3) 2- 3- -1;
(4) 2- . 3- 2-, -2;
(5) 2- . 2- 3- ;
(6) 2- . 1- 2-, -2, 3- 2-, -3;
(7) 3- , 3- . 3- 29;
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(8) 3- . 1- 3-, , -18, 2- 3-, 8.
, . :
1 |
+ 0 + 0z = 2
0 + 1 + 0z = 1.
0 + 0y + 1z = 0
= 2; = 1; z = 0.
:
, -1, ×-1 = -1× = , , .
Det ≠ 0, -1,
Det A = 0, -1 .
-1 = (1/det )×(٧),
Av- , :
11 121n
٧ = 21 222n
....
n1 n2nm
(1) :
× X = (5)
, , - ;
X .
det ≠ 0, -1. (5) -1 :
-1×× = -1× .
-1 × = , × = X,
X:
X = -1× . (6)
, det ≠ 0, , .
2 + 3 z = 8
: 2y z = 3
x 2y + 3z = -3
2 3 -1 8
: 0 2 -1 × = 3
1 -2 3 z -3
× = . = -1×,
2 3 -1 11 12 13
-1=(1/det )×(٧), det = 0 2 -1 = 7, ٧ = 21 22 23 .
1 -2 3 31 32 33
det ≠ 0, , . :
11 = 4; 12 = -1; 13 = -2; 21 = -7; 22 = 7; 23 = 7; 31 = -1; 32 = 2; 33 = 4;
4 -1 -2 4 -7 -1 4 -7 -1
٧ = -7 7 7; (٧)= -1 7 2; -1=1/7 -1 7 2;
-1 2 4 -2 7 4 -2 7 4
4 -7 -1 8 32 21 + 3 2
= 1/7 -1 7 2 × 3 = 1/7 -8 + 21 6 = 1.
-2 7 4 -3 -16 +21 12 -1
: = 2, = 1, z = -1.
.
= , -1. , -1 . :
X = -1 .
Y× = , -1. -1 . Y:
Y = B×C-1.
×× = -1 -1 . :
X = -1 -1.
1 -3 -1 -2 -9 2
: 2 2 0 × = 4 8 -2
2 4 2 12 16 -2
. ×=. =-1×.
1 -3 -1 11 12 13
-1 = (1/det )×(٧), det A = 2 2 0 = 12, ٧= 21 22 23 ;
2 4 2 31 32 33
4 -4 4 4 2 2 4 2 2 2 1 1
٧ = 2 4 -10; (٧) = -4 4 -2; -1 = 1/12 -4 4 -2 = 1/6 -2 2 -1;
2 -2 8 4 -10 8 4 -10 8 2 -5 4
2 1 1 -2 -9 2 12 6 0 2 1 0
= 1/6 -2 2 -1 × 4 8 -2 = 1/6 0 18 -6 = 0 3 -1.
2 -5 4 12 16 -2 24 6 6 4 1 1
.
, detA=0.
, ,
det A = 0, (),
.
( det = 0) j = j
( ). , -1 . .
+ + 3z = 5
: 2 2 + z = -3.
+ 5 + 8z = 10
1 1 3
. = 2 -2 1 = 0.
1 5 8
= 0, . :
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1 1 3 5 1 1 3 5 1 1 3 5
2 -2 1 -3 (1) 0 -4 -5 -13 (2) 0 -4 -5 -13;
1 5 8 10 0 4 5 5 0 0 0 -4
(1) 1- . 2- 1-, -2, 3- 1-, -1;
(2) 2- . 3- 2-. , 3- : 0 + 0 + 0z = -4.
x, y, z. , ().
+ 3z = 4
: 3 z = 16.
2 + 3z = 7
1 1 -3
. = 3 -1 -1 = 0.
1 -2 3
= 0, . :
1 1 -3 4 1 1 -3 4 1 1 -3 4 1 0 -1 5
3 -1 -1 16 (1) 0 -4 8 4 (2) 0 1 -2 -1 (3) 0 1 -2 -1;
1 -2 3 7 0 -3 6 3 0 1 -2 -1 0 0 0 0
(1) 1- . 2- 1-, -3, 3- 1-, -1;
(2) 2- -4, 3- -3;
(3) 2- . 1- 3- 2-, -1.
:
+ 0 z = 5
0 + 2z = -1
0 + 0 + 0z = 0.
, , z, . , . :
= z +5
y = 2z 1.
z = t, t R ( ). :
= t +5
y = 2t 1
z = t.
:
(t +5) + (2t 1) 3t = 4
3(t +5) (2t 1) t = 16
(t +5) 2(2t 1) + 3t = 7.
t, :
= 5 = 6
, t = 0 y = 1, t = 1 y = 1 . .
z = 0 z = 1