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2012ISBN: 978-960-337-110-6
Copyright 2012 -, , ( 3 . 2121/1993).
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, 1956 , . , . . 1981 2000 - . -.
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. , . (1970-2003) . , . (1958) /, . (1959-1966) /, / . ., . .
(1967-1969) , / . ., . (1967-1969) , . (1969-1971) - ., / . ., .
(1971-1972) .., / . ., . (1972-1974) .., / . ., . (1974-1977) - ..,
/ . ., . (1977) .., / . ., . (1977-1979) .., / . ., . (1979-1981) .
., / . ., . (1981-1982) .., / . ., . (1982-1984) .., / . ., . .
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1.1 . . .
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1.3 .
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1.1 . . .
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1 : 8000 , 5000 6000 .
2 : 6000 , 4000 2000 .
3 : 3000 , 2000 3000 .
4 : 2000 , 1000 1000 .
, - , - :
1 8000 5000 6000
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j
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zywx
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213
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11, 22, 33,..., .
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15
11
22
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500010003
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11 = 22= = 33=...= =1,
1 0 0 . 0
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200560065
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3502001
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yx ,
25020170000001
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11 12 1 1
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1
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i
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16
= [ji] .
, 1 5 6
2 4 2A
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264521
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() = .) A = [ij]
ij = ji i = 1,2,..., j = 1,2,..., . ,
352
512
2
yyx
x
3. - - , = .
1.1.1.
---. ( ..) - -, -, - 80, 20 10 .. . , . - .
.
, :
0 80
80 0 20
20 0 10
10 0
( - ) :
0 80 100 110
80 0 20 30
100 20 0 10
110 30 10 0
=
A .
4, - . ij = ji i = 1, 2, 3, 4 j = 1, 2, 3, 4 = .
-
17
1.1.2.
= [ij]23, ij = i j, i = 1, 2, j = 1, 2, 3) .) AT. ;) x
=
701212
3
2
xxxx
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.)
a a a11 12 131 1 0 , 1 2 1, 1 3 2= = = = = = ,
a a a21 22 132 1 1, 2 2 0 , 2 3 1= = = = = =
=
101210
A .
)
=
120110
TA .
, -.
)
=
101210
701212
3
2
xxxx
x 2 = 0, x2 2 = 2, x 1 = 1, x3 7 = 1.
x = 2 .
.
1.1.1. () ()() :
3523
531
2
2
zz
y
x
yx
301
+
200110812xy
[10]
[0 0 0 0 0]
365024001
100020001
101020001
-
18
1.1.2. 4 4 = [ij], ij = 2i 3j i = 1, 2, 3, 4 j = 1, 2, 3, 4. ,
.
1.1.3. 1.1 - 1, 2, . - - .. 1 3 . - - 1.1.
1.1.4. 1, 2, 3, 4, 5 12 - 1, 2,..., 12, 1.1. - = [ij]44 , aij i
j (.. a13 2, 1 3).
1.1.5. x, y, -
2
4 1 3 1 7 31 02 1 8 3 0 4
( ) ( )( )+ =
y x y x
x x x.
1.1.6. x 2
2
1 4
5 6 2
+
x
x x .
1.1.7. x, y,
2
2
2 2
1 1 1
1 0
0 1 3( )
=
+
x x
A x x
y x
. - =3.
1.1.8. 3
[ ]ij
a x y
A a x zy z
'' '
.
= [ij]
1 1 2 3 1 2 32
( ), , , = , , ij ij ji i j= + = ,
24
5
1 9
8
101112
6
7
2
3
4
5
3
1
. 1.1.
1
2
3
2
5
1
1
. 1.1.
-
19
. ;
1.2 .
, , , :
8 5 6
6 4 2
3 2 3
2 1 1
=
A
10 5 5
9 2 3.
7 8 1
1 5 2
=
B
, , , ( -) , ,
10 11
6 5
10 4
6 3
8 10 5 5 6 5 18
6 9 4 2 2 3 15
3 7 2 8 3 1 10
2 1 1 5 1 2 3
+ + +
+ + + =
+ + + + + +
.
. -:
, = [ij] B = [ij] , , - . + , + = [ij + ij] .
, . , ,
, - , .
2 0 1
3 2 1
4 6 2
1 4 1
8 10 5 5 6 5
6 9 4 2 2 3
3 7 2 8 3 1
2 1 1 5 1 2
=
.
( ). - :
-
20
= [ij] B = [ij] , , - . , = [ij ij] .
,
= + ().
, .
,
3 5 78 6 4
=
A , 2 4 64 6 8
=
B
1 42 53 6
=
,
3 5 7 2 4 6 5 9 138 6 4 4 6 8 12 12 12
+ = + =
A B ,
3 5 7 2 4 6 1 1 18 6 4 4 6 8 4 0 4
( )
= + = + = A B A B ,
3 5 7 1 2 3 4 7 108 6 4 4 5 6 12 11 10
TA
+ = + =
1 4 3 8 2 42 5 5 6 3 13 6 7 4 4 2
( )
= + = =
T T A A ,
, , , .
. , , , - ( ), :
1. + = + ( )
2. + = + = ( )
3. + ( + ) = ( + ) + ( )
4. + () = () + = ( )
+ ( + ) , ( + ) + , + + . , , , , , ( + ( + )) + , -
-
21
(( + ) + ) + , (A + B) + ( + ,) + (( + ) + ), + ( + ( + )),
, , , + + + . k (A1, A2,..., Ak, k 3 ),
1 + 2 + ...+ k.
+ = = .
+ = , + = , = . = , + = + , + = .
, , - :
5. ( + ) = + ( )
6. ( ) = ( )
1.2.1.
, - (15 ) .
1 13 11 2 1 0 3
2 10 9 2 2 3 4
3 8 6 4 4 3 5
) , , , , .
) 3 , 2 1 , : . .
) , , , . ( ) , .
.) , , , ,
-
22
:13 11 2 1 0 310 9 2 2 3 48 6 4 4 3 5
, ,A B = = =
.
) , = 3 + 2 + ,
13 11 2 1 0 3 39 33 4 2 0 3 43 383 10 9 2 2 2 3 4 30 27 4 4 3 4
37 35
8 6 4 4 3 5 24 18 8 8 3 5 35 31
= + + = + + =
X .
1 -- , 2 - .
1
433735
=
X , 2
383531
=
X .
)
13 2 0 11 1 310 2 3 9 2 48 4 3 6 4 5
,
= =
( ) ,
13 2 0 11 1 3 13 11 2 1 0 3 24 3 310 2 3 9 2 4 10 9 2 2 3 4 19 4
78 4 3 6 4 5 8 6 4 4 3 5 14 8 8
+ + + + = + = + + + = + + +
E .
- . , 2. , -
16 10 1212 8 46 4 64 2 2
2 8 2 5 2 6
2 6 2 4 2 2
2 3 2 2 2 3
2 2 2 1 2 1
=
.
2 . :
-
23
- . , , = [ij] = [ij] .
, , , , :
1. ( + ) = + 2. () = ()
3. ( + ) = +
4. lA = Al = A
5. = = 0 =
1.2.2.
0 3 42 2 54 1 1
=
A . , 5 3 = 2 3.
( , , . ).
. (,
) , .
3 3 3 315 2 3 5 2 3 3 3 33( ) = = = = X I X A X X I A X I A X I
A
:
1 0 0 0 3 4 1 0 0 0 9 12 1 3 9 121 10 1 0 3 2 2 5 0 1 0 6 6 15 6
19 3 153 3
0 0 1 4 1 1 0 0 1 12 3 3 12 3 10 3
/
/ .
/
X
= = + =
.
1.2.1.
1 2 0 1 2 3 4 02 3 3
2 0 1 2 2 0 3 10 1 2
0 1 1 3 0 3 2 5, , ,
= = = =
.
-, .
) , ) , ) , ) , ) , ) , .
-
24
1.2.2. 4 3 1 4 2 41 2 3 2 5 1
, ,A B
= = = ,
( ) + (), ( + ), + ( + ), ( + ) + , ( ) + . ;
1.2.3. 1 2 0 1 2 3 1 1 12 3 1 2 2 0 2 2 10 1 1 3 0 3 2 2 1
, ,A B
= = =
3 2 2 2 31 1, , , , ( )5 4
B A A B A + .
1.2.4. , . , , , DVD, - :
A A
20 30 33 25 18 75
DVD 55 40 39 15 23 29
. 29 15 48 15 50 18
90 80 65 30 66 85
) , .
) - .
) .
) ( ) , - .
1.2.5. , ( ) 1.2.4, :
A
400 380 390
DVD 80 100 90
. 290 250 310
90 120 100
-
25
) .
) 10%, - .
1.2.6. 1, 2, 3, 1 2.
1, 2, 3 1, 2, 31 2 3 1 2 3
1 230 40 50 36 48 6020 10 30 30 10 32
E E
= = .
) 1 212( )+E E .
) 20%, . - ( ) .
1.2.7. , 5 2 = 6 + 3
1 2 3 41 3 2 20 1 4 1
,
= =
A B .
1.2.8. ,
2 + 3 = 44 3 + 2 = 34 +
= [ij] 4 , 1, 2, 3, 4 1, 2, 3, 4.ija i j i j= = =
1.2.9. 3 0 4
23 1 2
A
=
1 1 38 2 4 4
1 1 1B
,
1 1 16 12 22 3 4
( )X B X A B = +
.
1.3 . 1.1. ,
8 5 6
6 4 2
3 2 3
2 1 1
=
A
( ) , , , - . ( ),
-
26
. :
) 5 , 6 8 .
) 2 , 1 3 .
, ( , - )
5 26 18 3
=
B .
1 , :
1 -
1
+
1 -
-
1
+
+
1
-
1
( )
8 ^ 5 + 5 ^ 6 + 6 ^ 8 = 118
1 1 .
, 1 - 2
8 ^ 2 + 5 ^ 1 + 6 ^ 3 = 39
1 .
( ) ( ) 2, 3 4 . , :
) 1 : 118 ( ), 39 ( ).) 2 : 70 ( ), 39 ( ).) 3 : 51 ( ), 17 (
).
-
27
) 4 : 24 ( ), 8 ( ). :
1 118 39
2 70 22
3 51 17
4 24 8
,
118 3970 2251 1724 8
=
.
. 11 1 1 -, 12 1 2 ....
:
= [ij] = [ij] , , , ij i j . , 1 1 3 3 2 2 + ...ij i j i j i j i
j = + + + j.
.
1 2 3
... ... ... ... ...
... ... ... ... ...
...
... ... ... ... ...i i i i a a a
1
2
3
... ...
... ...
... ...
... ... ...
... ...
j
j
j
j
... ... ...
... ... ...
... ...
... ... ...ij
=
j
i
- = , - (. 1.3).
. 1.3
-
28
1.3.1.
2.
=
, E 0 11 0
=
E .
, .
.
0 11 0
= =
AE
,
0 11 0
= =
EA
, , , . 2 ( ).
, , , ( - ), :
1. (kA) (k) = (kk) (A)
2. () = () ( )
3. ( + ) = + , ( + ) = + ( )
4. = = ( )
() , (), . , , , , (()), - (()), ()(), (()), (()), , , , .
k 1, 2, 3,..., k k, 1 2 3... k. A1 = 2 = 3 = ... = Ak = A, - 1 2
3... k = AAA ... A
k.
Al = A, A0 = I.
I k = I k.
k,r , :
5. A k A r = Ak+r
6. (Ak)r = Akr
7. ()k = kAk, R.
-
29
, :
8. (AB)T = BTAT.
, = , , , . , , , :
) . :
1 2 1 2 5 1 72 1 1 7 1 3 45 2 1 3 2 1 2
,
= =
A B ,
( 34), ( 34 33).
) , . , :
1 41 2 3
2 54 5 6
3 6,
= =
A B
. - 22, 33 =.
) , .
1 2 5 23 2 3 1
,
= = A B
, 1 2 5 2 11 4 5 2 1 2 11 63 2 3 1 9 4 3 1 3 2 6 4
,
= = = = AB BA .
) , . ,
1 2 5 32 1 3 5
,
= =
A B
:1 2 5 3 11 13 5 3 1 2 11 132 1 3 5 13 11 3 5 2 1 13 11
,
= = = =
AB BA
=. . -
. - :
-
30
) A , - .
) H = 0, = 0 = 0 , ..
1 10 0
=
A 0 10 1
=
B
0 00 0
= =
AB BA ,
= = . , (. = ), , , .
1.3.2.
, , ( + )2, ( + )3. , = . ( + )2, ( + )3.
. :
2 2 2( ) ( )( ) ( ) ( )+ = + + = + + + = + + +A B A B A B A B A
A B B A BA AB B
3 2 2 2( ) ( ) ( ) ( )( )+ = + + = + + + + =A B A B A B A BA AB
B A B
3 2 2 2 2 3= + + + + + + +A A B BA BAB ABA AB B A B .
(. =), .
=,
2 2 2 2 2 2 22( )+ = + + + = + + + = + +A B A BA AB B A AB AB B
A AB B
.
,
2 2
2
2
2 2
( ) ( ) ( ) ( ) ( ) ,
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
= = = = = =
= = = =
= = = =
= = = = = =
BA BA A AB A A BA A AB AA B A B
BAB BA B AB B A BB AB
ABA A BA A AB AA B A B
B A BA A AB A A BA A AB AA B A B
-
31
3 3 2 2 2 2 3
3 2 2 2 2 2 2 3
3 2 2 33 3
( )+ = + + + + + + + =
= + + + + + + + =
= + + +
A B A A B BA BAB ABA AB B A B
A A B A B AB A B AB AB B
A AB AB B
- .
( + )2, ( + )3 = = .
2 2 2 22 2( ) .+ = + + = + +A I A AI I A A I3 3 2 2 3 23 3 3 3(
)+ = + + + = + + +A I A A I AI I A A A I .
1.3.3.
n- (n )
11
22
0 00 0
0 0 0
...
...
... ... ...
=
aa
A
a
n- ,
11
22
0 0
0 0
0 0 0
...
...
... ... ...
=
n
nn
n
a
aA
a
.
. . n = 1
. n = k,
11
22
0 0
0 0
0 0 0
...
...
... ... ...
=
k
kk
k
a
aA
a
.
n = k+1, Ak+1 1
111
22
1
0 0
0 0
0 0 0
...
...
... ... ...
+
+
+
k
k
k
a
a
a
-
32
11
1 22
0 0
0 0
0 0 0
...
...
... ... ...+
= =
k
kk k
k
a
aA A A
a
11
22
0 00 0
0 0 0
...
...
... ... ...
aa
a
:
111
11 22
1
0 0
0 0
0 0 0
...
...
... ... ...
+
++
+
=
k
kk
k
a
aA
a
.
, .
.
1.3.1.
1 2 0 1 2 3 4 02 3 3
2 0 1 2 2 0 3 10 1 2
0 1 1 3 0 3 2 5, , ,A B
= = = =
.
, , , , , , , , , .
1.3.2.
1 2 1 2 5 1 7 3 6 0 62 1 3 2 1 3 4 1 2 4 55 2 3 3 2 1 2 4 3 2
3
, ,A
= = =
.
) , , , .) , .) ();
1.3.3.
3 2 0 1 02 1 2
1 0 2 0 11 2 1
0 1 3 2 0, , .A B
= = =
: + 2 + 3, 32 2, , 2 3:
1.3.4. M 1 2, - 1, 2 3. 1, 2, 3 - .
1 2 3 1 2
1 1 5 2 51 2 31 5 2 4
1
2
3
, ,
,
A
10 1615 1713 14
1
2
3
B
-
33
) .)
2 1, 3 2 5 3 ;
1.3.5. : -. - .
1 60 752 30 603 10 70
20 50 15 40
, .
1.3.6. .
1 0 0 0 1 2 01 0 0 2 2 2
0 1 0 0 2 0 2 3
=
1.3.7. = [x y z], B = [ ] =. - x2 + y2 + z2;
1.3.8.
=
x yA
y x
1 =
B
, , , x, y, R y 0,
AB = BA a = 1.
1.3.9.
=
x yA
z w A1 (x + w)A + (xw yz)I2 = .
1.3.10. 3 11 2
=
A , A2 + 5A -
2.
1.3.11. A
00
0
A
=
2 + 2 + 2 = 1, A3= A.
1.3.12. , ( + )2 = 2 + 2 .
1.3.13. , ( + 3)(2 + ) = (2 + )( + 3) - .
-
34
1.3.14. , = = .
1.3.15.
0 1 13 2 32 2 3
=
A
4 3 32 1 23 3 2
=
B . -
A2 = B2 = I3, AB + BA =2I3, (A B)2=O. : -
A2=B2, A=B A= B;
1.4 .
a a 0 ,
1a a
1, aa1=a1a=1. : , AB=BA=I;.
AB BA . :
. - AB = BA = I, .
, . , , A1. :
AA1 = A1A = I.
, 1 2 5 23 5 3 1
,
= = A B
21 2 5 2 1 03 5 3 1 0 1
= = =
AB I
25 2 1 2 1 03 1 3 5 0 1
= = =
BA I .
, . ,
. .
, - AB=I BA=I, A,B , .
, , - :
-
35
, - ,
AX = B X = A1 B XA = B X = BA1. (1.4.1)
: ) AX = B, , 1 1 1( )=( ) = = =A B A AX A A X IX X .) X = A1B,
, 1 1( ) ( ) = = = =AX A A B AA B IB B . . ,
(. AB=O), , . AB=O , ,
. , , :
1 1 = = = =O O O OAB A AB A IB B .
, 2 - . .
) O
=
A
, , D = 0.
)
=
A
, ,
11 =
A
D. (1.4.2)
.
A
=
2. , -
x y
Xz
=
= ,
1 0 1 00 1 0 1
x y x z y
z x z y
+ + = = + +
x, y, z,
10
x zx z
+ = + =
(1) 01
y y + =
+ = (2).
, (1) (2) . D = :
) D 0, (x, z) xD
=
zD
= , -
(1), (y, )
yD
=
D
=
-
36 (2).
D DX D D
=
11
A D
= .
-
D DX D D
=
(
D 0) = = .
) D =0, (1) (2) , . :
0 = 0 0 0, (1) (2) .
= = = = 0, . ,
1 23 5
=
A , D = 1 5 2 3 = 5 6 = 1 0
1 5 2 5 2 5 21 13 1 3 1 3 11
( )
= = = A .
, 1 23 6
=
A , D = 1 6 2 3 = 6 6 = 0.
1.4.1.
, ,
()1 = 11.
. () (1 1) = (
1 1) () = . ,
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
( )( ) (( ) ) ( ( )) ( ) ,
( ) ( ) (( ) ) ( ( )) ( ) .
AB B A AB B A A BB A AI A AA I
B A AB B A A B B A A B B I B B B I
= = = = =
= = = = =
1.4.2.
0 1 1 4 3 33 2 3 2 1 22 2 3 3 3 2
,
= =
A B .
-
37
) 2, 2.) .) = =.
.)
23
0 1 1 0 1 1 1 0 03 2 3 3 2 3 0 1 02 2 3 2 2 3 0 0 1
= = =
A I
2
3
4 3 3 4 3 3 1 0 02 1 2 2 1 2 0 1 03 3 2 3 3 2 0 0 1
= = =
B I
) 2 = 2 = 3 = = 3 = = 3 1 = 1 = .
) 1 = , :
10 1 1 4 3 3 5 4 4 5 4 43 2 3 2 1 2 1 2 1 1 2 12 2 3 3 3 2 5 5 4
5 5 4
.AX B X A B X AB X
= = = = = =
10 1 1 4 3 3 5 4 4 5 4 43 2 3 2 1 2 1 2 1 1 2 12 2 3 3 3 2 5 5 4
5 5 4
.AX B X A B X AB X
= = = = = =
, 1 = , :
14 3 3 0 1 1 3 4 42 1 2 3 2 3 1 0 13 3 2 2 2 3 5 5 6
= = = = =
BY A X B A X BA .
.
1.4.1. , , 2.
1 2 3 41 2 6 3 6 34 3 4 2 4 2
, , ,
= = = =
A A A A
.
1.4.2. 1 02 1
=
A 4 73 5
=
B .
) .) 1 = .
) 1 = .
-
38
1.4.3. , - .
)
1 2 3 0 1 114 1 1 3 2 113
1 1 1 3 1 7,
= =
A B )
2 1 2 1 1 21 1 0 1 2 22 0 1 2 2 3
,
= =
A B
)
1 1 1 1 15 8 1 4 1 4 3 81 1 2 3 1 4 1 2 1 2 1 41 2 1 3 1 4 1 2 1
2 1 41 3 3 1 3 8 1 4 1 4 1 8
/ / / /
/ / / /,
/ / / /
/ / / /
= =
A B
1.4.4.
21 3 90 1 60 0 1
=
x x
A x .
)
21 3 90 1 60 0 1
=
x x
B x .
)
1 6 360 1 120 0 1
.
1.4.5. 3 42 3
=
A 1 20 1
=
B .
) .
) (AB)1 .
1.4.6. x, y, z , A,B 3 .
2 1 11 2 11 1 1
=
A ,
0 13
3 1
=
x
B y x
z
.
1.5 .
A = [aij] , B = [ij] .
1 2 1 2( , ,..., , ,..., )ij ijA B a i j = = = =
. , .
.
11
22
0 00 0
0 0 0
...
...
... ... ... ...
aa
a
-
39
.
1 0 0 00 1 0 00 0 1 0
00 0 0 1
...
...
...
... ... ... ...
...
= =
I I
. , .
. ij = ji i = 1, 2, ..., j = 1, 2, ..., = A.
A = [aij] = [ij] , : , , R.
[ ]
[ ]
[ ]
ij ij
ij ij
ij
A B a
A B a
A a
+ = +
=
=
A = [aij] = [ij] . 1 1 2 2
[ ] ,
+...
ij
ij i j i j i j
AB
a a a
=
= + +
.
1 = 1 =
2
=
A
D = 0
1 1 =
A
1.6 .
, , .
1. A = [aij] + .
2. (2 + 2) = , = .
3. A 3( ) = , =.
4. , 34, .
5.
0 0 00 1 00 0 0
.
-
40
6. 2 = , = = .
7. = , = .
8. = 5, , , , .
9. ( )2 = , , A = I.
10. , , ()2 = 22.
11. 3 22 1
3 22 1
.
12. = , , .
13. = .
14. , = , = 1.
15. , , .
.
1. , , , :
) , 1 ) 1, 1 ) 1, )
2.
1 1
2 2
a a
1
0
:
)
1
2
)
1
1
a
)
1
2
a
a
)
1
2
a
3. , 410, :
) . ) . ) . ) .
4.
2
2
2
3 3 0 0
0 1 0
0 0 4 4
+
+
+
a a
a a
a a
:
) = 1. ) = 2. ) = 0. ) .
-
41
5. = =, :
) = ) = ) = ) =
6.
x 0 . ' - x :
) xA1 ) 1A
x)
11 Ax
) xA
7.
,, -, :
) ( + ) = + . ) ( ) = ().
) ( + ) = + . ) .
8.
1 42 8
:
)
3 22 1
.
) 3 22 1
.
) .
)
3 22 1
.
9.
1 0 04 2 06 5 3
:
) . ) . ) . ) .
10.
2 5 4 00 3 5 20 0 1 30 0 0 0
:
) . ) . ) . ) .
11.
1 0 04 2 06 5 3
:
) . ) . ) . ) .
12.
, , , :) + ( + ) = ( + ) + .
) + = + .
) ( ) = ( ) + .
) .
-
42
13.
, , : ) 5(A + B)= 5A + B
) = = 0 =
) =
) ( + )2 = 2 + 2 + 2
1.7 .
1.7.1. 2 2 21 2 ...+ + + x x x - = [x1, x2, ..., x].
1.7.2. :)
.) -
.) -
.
1.7.3. . :) . ) .
1.7.4. , . :) . ) + .
1.7.5. = . - , . :) .) + .
-
x
x
limx x0
2
2.1 .
2.2 .
2.3 I .
2.4 .
2.5 .
2.6 .
2.7 .
2.8 .
2.9 .
2.10 .
2.11 .
, , - ... - . , . - - , , , . Cramer , .
-
44
2.1 .
A x1, x2
11 1 12 2 1
21 1 22 2 2 .
x x
x x
+ =+ =
()
11 1 12 2 1
221 1 22 2
x x
x x
+ =
+ .
, 22 11 1221 22
A
=
(-
) 21 12
xX
x
=
( -
).
11 12 1 1
21 22 2 2
x AX B
x
= =
(2.1.1)
1
2
B
=
.
, 22 12, :
11 22 1 12 22 2 22 1
21 12 1 22 12 2 12 2
x x x x
,
11 22 21 12 1 22 1 12 2( ) x . (2.1.2) ,
D = 11 22 21 12 0
x1
22 1 12 2 22 1 12 21
11 22 21 12
x
D
= =
. (2.1.3)
x2
11 22 21 12 2 11 2 21 1( ) a x (2.1.4)
, D 0,
11 2 21 1 11 2 21 12
11 22 21 12
x
D
= =
. (2.1.5)
D = 11 21 21 22 ,
-
45
11 12
21 22
A
=
2 ( -
2 2 ) 11 12
21 22
.
:
2
11 12
21 22
A
=
.
,
11 1211 22 21 12
21 22
a aA a a a a
a a= = . (2.1.6)
:
4 35 2
A
=
4 34 2 5 3 7
5 2 | | ( ) ( )A
= = =
,
1 00 1
I
=
1 0
1 0 10 1
|I|= = =
0 00 0
=
O 0 0
0 0 00 0
= = =O .
(2.1.3), (2.1.5) - . ,
Dx1 1 21 12 11 1
1 22 2 12 11 2 21 12 22 21 2
, x x
D D
Dx2 1 21 12 11 1
1 22 2 12 11 2 21 12 22 21 2
, x x
D D
.
, D 0 ,
1 2
1 12 11 1
2 22 21 21 2
11 12 11 12
21 22 21 22
,x x
D D
x x D D
Dx1D
1 2
1 12 11 1
2 22 21 21 2
11 12 11 12
21 22 21 22
,x x
D D
x x D D
Dx2D
1 2
1 12 11 1
2 22 21 21 2
11 12 11 12
21 22 21 22
,x x
D D
x x D D
(2.1.7)
( Cramer). Dx1, Dx2 D =
x1, x2 . () (2.1.2), (2.1.4),
D ^ x1 = Dx1, D ^ x2 = Dx2
:
-
46
11 1 12 2 1
21 1 22 2 2 .
x x
x x
+ = + =
11 12
21 22
D
= -
.
D 0
1 2
1 12 11 1
2 22 21 21 2
11 12 11 12
21 22 21 22
,x x
D D
x x D D
Dx1D
1 2
1 12 11 1
2 22 21 21 2
11 12 11 12
21 22 21 22
,x x
D D
x x D D
1 2
1 12 11 1
2 22 21 21 2
11 12 11 12
21 22 21 22
,x x
D D
x x D D
Dx2D
1 2
1 12 11 1
2 22 21 21 2
11 12 11 12
21 22 21 22
,x x
D D
x x D D
. (2.1.8)
D = 0 Dx1 0 Dx2 0, -
. D = Dx1
= Dx2 = 0, ,
11 = 21 = 12 = 22 = 0 1 0 2 0,
.
2 3 , 33 .
11 12 13
21 22 23
31 32 33
a a a
A a a a
a a a
=
.
22 23 21 23 21 2211 12 1332 33 31 33 31 32
+
11 12 13
21 22 23
31 32 33
.
:
3
11 12 13
21 22 23
31 32 33
a a a
A a a a
a a a
=
.
, :
11 12 1322 23 21 23 21 22
21 22 23 11 12 1332 33 31 33 31 32
31 32 33
a a aa a a a a a
A a a a a a aa a a a a a
a a a
= = + .
-
47
, - , () . () ij 2 , ij . (1)
i+j. 2 ,
11 22 33 32 23 12 21 33 31 23 13 21 32 31 22( ) ( ) ( )A a a a a
a a a a a a a a a a a= +
21 12 33 32 13 22 11 33 31 13 23 11 32 31 12
12 13 11 13 11 1221 22 23
32 33 31 33 31 32
( ) ( ) ( )
.
A a a a a a a a a a a a a a a a
a a a a a aa a a
a a a a a a
= +
= +
-.
,
22 23 12 13 12 1311 21 31
32 33 32 33 22 23
a a a a A a a a
a a a a .
., () -
, .
1 1 1 2 1 3
2 1 2 2 2 3
3 1 3 2 3 3
1 1 1
1 1 1
1 1 1
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
+ + +
+ + +
+ + +
+ + + = + +
.
,
1 2 13 2 52 0 0
A
= ,
, , :
2 1 1 1 1 22 0 0 2 10 2 0 5 3 0 2 6 16
2 5 3 5 3 2( ) ( ) ( ) .A
= + = + + + =
, , - .
-
48
33
11 12 13
21 22 23 11 22 33 32 23 12 21 33 31 23 13 21 32 31 22
31 32 33
( ) ( ) ( )
A
11 22 33 12 31 23 13 21 32 11 32 23 12 21 33 13 31 22
, Sarrus. - (+) () - - (. 2.1).
, ( Sarrus ).
3231333231
2221232221
1211131211
DDDDDDDDDDDDDDD
. 2.1. Sarrus.
+ + +
2.1.1.
2 2 31 2 12 2 1
A
=
.
) =2 .) .
.
) =2
0 2 31 0 12 2 1
A
=
-
0 2 30 1 2 3 2 3
1 0 1 0 1 22 1 2 1 0 1
2 2 1
2 1 3 2 2 2 1 0 3 4 4 0 [ ( ) ( )] [ ] .
A = = + =
= + = + =
)
-
49
2 2 32 1 1 1 1 2
1 2 1 2 2 32 1 2 1 2 2
2 2 1( )
A
= = + =
2 2 1 2 1 2 1 1 1 2 3 1 2 2 1( )[( ) ( ) ( ) ] [ ( ) ] [( ( ) (
)] = + =3 25 2 8. = +
=2, , - .
y
xO
B(x2 , y2)
(x1 , y1)
y2
y1
(x2 - x1)
. 2.1.
y
xO
B(x2 , y2)
(x3 , y3)
(x1 , y1)
. 2.1.
y
xO
B(x2 , y2)
(x3 , y3)
(x1 , y1)
. 2.1.
2.1.2.
, , xOy - (x1, y1), (x2, y2), (x3, y3) , - (. 2.1)
3 3
2 2
1 1
11 121
( )
x y
x y
x y
.
. , , , ,
Ox, 2.1, ,
() = () + () ().
, (. 2.1) -
2
( )
ABE
-
A = y1, BE = y2, = x2 x1 :
1 2 2 112
( ) ( ) ( )ABE y y x x .
2 3 3 2
1 3 3 1
12 2
12 2
( ) ( ) ( )
( ) ( ) ( ).
y y x x
y y x x
,
1 2 2 1 2 3 3 2
1 3 3 1
1 12 212
( ) ( ) ( ) ( ) ( )
( ) ( )
AB y y x x y y x x
y y x x
-
50
1 2 1 3 2 3 2 1 3 2 3 1 1 2 3 2 3 1 3 1 21 12 2
( ) ( ) ( ) ( ) ( ) .AB y x y x y x y x y x y x y x x y x x y x
x
3 3
2 2
1 1
111
x y
x y
x y
3
3 32 3 3
2 2 3 2 1 3 1 2 2 1 3 1 2 31 1 2
1 1
11 1 1
11 1 1
1( ) ( ) ( )
x yx x x
x y y y y y x x y x x y x xx x x
x y
3 3
2 2
1 1
11 121
( ) .
x y
AB x y
x y
, , , , (.. , ... ), , - , . , . (4,5), (5,2) (1,2) -
1 4 51 11 5 2 12 62 2
1 1 2( ) AB = = = .
.
2.1.1. (2.1.8).
) 2 3 5
3 2 5
x y
x y
+ =
+ =
)
2 3 5
3 6 3
x y
x y
+ =
+ =
)
2 5 1
4 10 2
x y
x y
=
+ =
2.1.2. , 2 .
)
1 0 52 2 43 0 1
)
1 0 50 2 03 0 4
)
9 3 26 2 40 0 1
)
010
)
2 2
1 0 01
)
1 01 11 1 0
-
51
2.1.3.
)
1 1 21 1 01 3
x
x x
x x
=
)
2 1 11 1 2 03 3
x x
x x
x x
+ + =
) 2 3
05 2 3x x
x x=
+
)
2
2
2
1 11
1 11 1
1 1
xx
x x
x
=
2.1.4. , , , (1, 5), (2, 3), (2, 1) -. 2.1.2, .
2.1.5.
2 2 2
1 1 1( )( )( )
a
a a a= .
2.1.6.
1 1 1 1 11 1 1 11 1 1 1
a
a a a
a a
=
.
2.1.7. , - 0.
, , ,
a a a a
x y z x y z x y z a x y z
x y z a x y z x y z x y z
+ + ++ + +
.
2.1.8. , - .
,
x y z k m n
k m n x y z
2.1.9.
1 11 11 1
A
=
.
) =2 .
-
52
) .)
1 11 1 11 1
.
2.2 .
,
11 12 13
21 22 23
31 32 33
a a a
A a a a
a a a
=
3, - ,
12 13 11 13 11 1221 22 23
32 33 31 33 31 32
a a a a a aA a a a
a a a a a a= + ,
22 23 21 23 21 2211 12 13
32 33 31 33 31 32
a a a a a aA a a a
a a a a a a= + ,
22 23 12 13 12 1311 21 31
32 33 32 33 22 23
a a a a a aA a a a
a a a a a a= + .
, , ij - 2 Mij , ij. ij mij =Mij.
, + , (1)i+j. (1)i+j mij ij - cij,
1 1( ) ( )i j i jij ij ijc m M+ += = .
,
21 21 22 22 23 23A a c a c a c= + + ,
11 11 12 12 13 13A a c a c a c= + + , (2.2.1)
11 11 21 21 31 31A a c a c a c= + + .
,
31 31 32 32 33 33A a c a c a c= + + ,
12 12 22 22 32 32A a c a c a c= + + , (2.2.2)
-
53
13 13 23 23 33 33A a c a c a c= + + .
3 ( 33 ) 2 .
, -, A = [ij] . 3 1 ( ).
A = [ij] .
11 12 1
21 22 2
1 2
= 11 c11 + 12 c12 ++ 1 c1 cij = (1)i+j mij = (1)i+jMij Mij - 1 ,
ij.
3 , mij =Mij ij cij = (1)
i+j mij ij. 11 c11 + 12 c12 ++ 1 c1 , , 1 .
3 , 1. :
11 12 1
21 22 2
1 2
A
=
.
1 1 2 2 1 1 2 2, i i i i i i j j j j j jA a c a c a c A a c a c
a c= + + + = + + +
i j 1,2,,. i , , j .
, , - .
, - , = 0.
-
54
2.2.1.
0 00
0
a a a a
x a a x
a x a
a a a x
=
.
. 2 (
)
2 1 2 321 23 21 23
0 01 1
0( ) ( ) ( ) ( ) ( )( )
a a a a
x a a xx a M a x M x a M M
a x a
a a a x
+ + = + = +
21 230 0,a a a a a a
M x a M a a
a a x a a x
= = .
21 1 23 2 ,
1 1 1 3 2 2 221
1 2 3 2 2 223
0 1 1
0 1 1
( ) ( ) ( ) ( ) ( ),
( ) ( ) ( ) ( ).
a a ax a a a
M x a a a a x a a a ax ax x aa x x a
a a x
a a aa a a a
M a a a a a ax a a x aa x a a
a a x
+ +
+ +
= = + = + =
= = + = =
1 1 1 3 2 2 221
1 2 3 2 2 223
0 1 1
0 1 1
( ) ( ) ( ) ( ) ( ),
( ) ( ) ( ) ( ).
a a ax a a a
M x a a a a x a a a ax ax x aa x x a
a a x
a a aa a a a
M a a a a a ax a a x aa x a a
a a x
+ +
+ +
= = + = + =
= = + = =
2 20 000
( )[ ( ) ( )] ( ) ( )
a a x a
x a a xx a ax x a a x a a x a x a
a x a
a a x
= = +
, a(x a)2(x + a) x = a x = a.
-
55
2.2.2.
,
11
21 22
1 2
0 00
A
=
,
A
=
11 12 1
22 20
0 0
= 11 22 .
, - .
. 33
0 0
0 .
0
A
=
11
21 22
31 32
( 2.2.1) :
0 0
A
= + +
22 21 21 2211
32 33 31 33 31 32
0 0
0
( ) .
A
= = =
2211 11 22 33 32 11 22 32
32 33
0
- , .
2.2.3.
:
11 11 12 1
21 21 22 2
1 1 2
a
A
+ + = +
:
11 12 1 11 12 1
21 22 2 21 22 2
1 2 1 2
| | .
A
= +
.
-
56
.
2.2.1.
1 2 3 4 1 2 3 4 3 1 2 40 5 6 7 0 0 8 9 8 0 0 90 0 8 9 0 0 0 0 0
0 0 00 0 0 10 0 0 0 10 0 0 0 100 0 0 0 0 5 6 7 6 5 0 7
, , .
a a a
;
2.2.2.
1 0 0 0 1 0 0 0 4 0 0 0 1 0 0 01 2 0 0 1 2 0 0 1 3 0 0 11 2 0 01
2 3 0 1 2 0 0 1 2 2 0 111 22 3 01 2 3 4 1 2 3 4 1 2 3 1 1111 222 33
4
, , , .
. 1
11 11 12 1
21 21 22 2
1 1 2
++
=
+
v
a
A
( )11 11 11 21 21 21 1 1 1 = + + + + + + ( ) ( ) c c c
1 11 1 11 1( ) ( )
i ii i ic m M
i1, i = 1, 2, ... .
11 11 21 21 1 1 11 11 21 21 1 1( ) ( ) A c c c a c a c a c ...11
11 21 21 1 1 11 11 21 21 1 1( ) ( ) A c c c a c a c a c ...11 11 21
21 1 1 11 11 21 21 1 1( ) ( ) A c c c a c a c a c
11 12 1
21 22 2
1 2
1 ,
11 12 1
21 22 2
1 2
1 .
-
57
2.2.3.
22
22
2 22 2
2 2 2
1 0 0 0 01 0 0 0
1 2 0 0 01 2 0 0
1 3 0 0 01 3 0
1 1 4 01 1 4
4 1 1 5
xx
x xx x
x x xx x x
x x x xx x x x
x x x x x
+
+ + +
+ =+ +
+ + +
+
.
2.2.4.
1 2 3 4 5 1 1 1 1 11 1 1 1 1 1 1 1
0 1 2 3 4 0 111 1 1 10 1 1 1 0 1 1 1
0 0 1 2 3 0 0 0 111 10 0 1 1 0 0 1 1
0 0 0 1 2 0 0 0 111 10 0 0 1 0 0 0 1
0 0 0 0 1 0 0 0 0 1
, , , .
2.2.5.
1 0 1 11 0 1 11 1 0 11 1 0 1
x p x p x p
y q y q y q
w r w r w r
z s z s z s
= +
2.2.3, .
2.2.6.
1 1 0 1 0 1 0 11 0 1 1 0 1 0 11 0 1 0 1 1 0 11 0 1 0 1 0 1 1
x p x p x p x p x p
y q y q y q y q y q
w r w r w r w r w r
z s z s z s z s z s
= + + +
2.2.3, .
2.2.7. - 2.2.2 .
0 0 0
0 0 0 ,
0 0
0 0 0
11 12 13 14 14
21 22 23 23 24
31 32 32 33 34
41 41 42
0.
43 44
-
58
2.3 I .
, - , , , . 2, (2.1.6). , , , .
, .
2.2.2 :
D1 , .
, 2 3 4
24
1
0 21 1 2 3 4 5 1200 0 3 12
0 0 0 40 0 0 0 5
.
x
x
x x x x
e x x
xx
xe
= =+
D1 :
D2 , ,
11
2211 22
0 00 0
0 0
"" "# # #"
.
D3 1, = 1.
, - :
D4 ( ) , .
, , .
D5 ( ) , - .
D6 ( ) , .
-
59
( -).
D7 ( ) ,
=.D8
( ) ( ) , = .
D9 ( ) ' ( ) ( ) , =.
, , ' .
D10 , =.
D11 , =. = 2=2 , -
2 3, , ,...k kA A k= =
D12 , = .
D10, D11, D12 =2. ,
, x y
A B z
= =
,A B x yz= =
,Ta a
A A
= =
x y x z y
AB z x z y
+ + = = + +
.
) TA a a A= = = ,
) 2 2 ( )( ) ( )( ) ( )A A= = = ,
) ( )( ) ( )( )AB ax z y ay x z axy ax zy z= + + + + = + + +
( ) ( )yx yz x z x x yz zy =
( ) ( ) ( )( ) x yz x yz A B= = = .
, .
-
60
2.3.1.
1 3 52 4 69 7 10
A
=
-
, .
. (2)
1 3 50 2 49 7 10
D9 ,
1 3 5 1 3 52 4 6 0 2 49 7 10 9 7 10
A = = .
, (9) -
1 3 50 2 40 20 35
A =
.
(10)
1 3 50 2 40 0 5
A =
, D1 ,
1 3 50 2 4 1 2 5 100 0 5
( ) = = .
= 10.
2.3.2.
0,1,2 3 (x) = 0
2
3
4
1 1 1 2
1 1 3 8
1 1 7 26
1 1 15 80
( )
x
xA x
x
x
=
.
-
61
. x = 0, x = 1, x = 2 x = 3,
1 1 1 21 1 3 8
01 1 7 261 1 15 80
( )A
=
,
0 1 1 20 1 3 8
10 1 7 260 1 15 80
( )A = ,
0 1 1 20 1 3 8
20 1 7 260 1 15 80
( )A = ,
0 1 1 20 1 3 8
30 1 7 260 1 15 80
( )A =
. A(0) [ 2 1 (1)], , D6 A(0)=0. , 0 A(x)=0.
A(1) , ( D4 ) A(1)=0. , 1 A(x)=0. , A(2), A(3) , ( D5 -) A(2)=0
A(3)=0. 2 3 A(x)=0.
2.3.3.
5 5 5 55 5 5 55 5 5 55 5 5 55 5 5 5
x
x
A x
x
x
=
(. ).
. , , 1
20 5 5 5 520 5 5 520 5 5 520 5 5 520 5 5 5
x
x x
x x
x x
x x
+ + + + +
D9 ,
5 5 5 5 20 5 5 5 55 5 5 5 20 5 5 55 5 5 5 20 5 5 55 5 5 5 20 5 5
55 5 5 5 20 5 5 5
x x
x x x
A x x x
x x x
x x x
++
= = +++
.
-
62
D8,
20 5 5 5 5 1 5 5 5 520 5 5 5 1 5 5 520 5 5 5 20 1 5 5 520 5 5 5
1 5 5 520 5 5 5 1 5 5 5
( )
x
x x x
A x x x x
x x x
x x x
++
= + = +++
.
(1) , ,
1 5 5 5 5 1 5 5 5 51 5 5 5 0 5 0 0 0
20 1 5 5 5 20 0 0 5 0 01 5 5 5 0 0 0 5 01 5 5 5 0 0 0 0 5
( ) ( )
x x
A x x x x
x x
x x
= + = +
, D1 ,
4 4
1 5 5 5 50 5 0 0 00 0 5 0 0 1 5 50 0 0 5 00 0 0 0 5
( ) ( )
x
x x x
x
x
= =
.
= (x 5)4.
2.3.4.
, xOy (x1, y1), (x2, y2), , ,
1 1
2 2
11 01
x y
x y
x y
= . (2.3.1)
. 1
1 1 1 1
2 2 2 2
1 11 0
1 1x y y x
x yx y y x
+ =
x+y= 1 1 1 1
2 2 2 2
1 11 1
y x x y
y x x y= = = .
-
63
, (2.3.1) . , (2.3.1) - x=x1, y=y1 x=x2, y=y2,
1 1
1 1
2 2
11 01
x y
x y
x y
= , 2 2
1 1
2 2
11 01
x y
x y
x y
=
( D5). , ,
1 1
2 2
11 01
x y
x y
x y
=
.
, (1, 2) (2, 1)
11 2 1 2 1 1
1 1 2 0 1 0 32 1 1 1 1 2
1 2 1
x y
x y x y
= + = + =
.
2.3.5.
1 1 11 1 11 1 11 1 1
x p x p y q y q w r w r z s z s
++
=++
.
. 2.2.3,
1 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 1
x p x p x p y q y q y q w r w r w r z s z s z s
++
= +++
( ),
1 1 11 1 11 1 11 1 1
x p x p
y q y q
w r w r
z s z s
++
=++
.
-
64
.
2.3.1. 1 (0, 0) (5, 4). 2 - (3, 1) (4, 5).) 2.3.4,
1 1 1 2 2 2, x y x y + = + =
) (2.1.8), .
2.3.2. 2.2.2 D7 ( ).
2.3.3. D12 D5 ( , D6) .
1 1 1 2 21 1 2 3 41 2 3 4 3 3 4 4 41 4 2 3 10 1 0 1 1 2 3 4 52 5
6 6 21 0 1 0 1 1 2 2 22 3 3 6 35 10 15 20 2 2 3 3 3
3 1 1 9 1 1 2 3 4 5
, ,
2.3.4.
2 22 2 02 2
x x x
x
x
=
x = ( + ).
2.3.5. (. ) =1.
1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1
, , ,
.
2.3.6. 1, 2 3
2
3
1 1 2
1 1 4 0
1 1 8
( )
( )
x
x
x
=
.
2.3.7. , , .
4 3 3 4 3 4 2 2 3 4 3 4 23 2 6 6, ( ) ( ) , , ( ) ( ), , , , (
)T T T T T T T TA B A B A B A B B B B A B A B A
-
65
2.3.8. , , = 1, = 2, = 3,
3 2 2 23 3, , , , T TAB A B A B AB B A .
2.3.9. :
) = ) 2= )
3=
2.3.10.
2 1 11 2 11 1 2
x
A x
x
=
(. ).
2.3.11. ) ,
2
2
2
1
1
1
x x
A a a
=
.
)
2 2 21 1 11 1 1 3 9 1 3 3 9 12 4 1 2 4 1 4 16 1
x x x x x x
+ = .
2.3.12.
3 3 3 3
2 2 2 26
8 27 64
4 9 16 122 3 4
1 1 1 1
.
a a a a
a a a a aa a a a
=
2.3.13. , , .
0 1 1 2 21 1 1 1
2 3 4 2 3 4 5 62 2 3 3
3 4 2 1 2 3 2 12 2 2 2
4 2 3 6 5 4 3 21 2 3 4
2 2 1 1 0
, ,
2.3.14. xOy (x1, y1), (x2, y2), (x3, y3), . 2.3.4,
-
66
1 1
2 2
3 3
11 01
.
x y
x y
x y
=
2.3.15. ) ,
11
11
+ + + +
1++++.) 4,
2 2 3 4 3 3 4 51 3 3 4 2 4 4 51 2 4 4 2 3 5 51 2 3 5 2 3 4 6
A
=
.
.
2.3.16. ) ,
1 1 1 11 1 1 11 1 1 11 1 1 1
xxyz
y
z
+=
++
.
) 4,
1 1 1 1 1 1 1 11 2 1 1 1 4 1 11 1 3 1 1 1 3 11 1 1 4 1 1 1 2
A
=
.
.
2.3.17. D13 , - .
1 1 2 2 3 3 1 1 2 2 3 31 0 0 1 1
1 2 0 0 1 1 1 2 0 0 1 10 1 0 1 2
0 0 2 0 0 0 3 4 2 3 4 50 0 1 1 3
0 0 3 4 0 0 4 5 0 4 5 61 1 1 1 4
0 0 4 5 6 0 5 6 0 0 6 21 2 3 4 5
0 0 5 6 8 1 6 7 0 0 0 8
, ,
1 1 2 2 3 3 1 1 2 2 3 31 0 0 1 1
1 2 0 0 1 1 1 2 0 0 1 10 1 0 1 2
0 0 2 0 0 0 3 4 2 3 4 50 0 1 1 3
0 0 3 4 0 0 4 5 0 4 5 61 1 1 1 4
0 0 4 5 6 0 5 6 0 0 6 21 2 3 4 5
0 0 5 6 8 1 6 7 0 0 0 8
, ,
-
67
2.4 .
3
11 12 13
21 22 23
31 32 33
a a a
A a a a
a a a
=
mij = ij aij cji = (1)i+j mji = (1)i+jij aij i = 1, 2, 3 j = 1,
2, 3. C ,
11 12 13
21 22 23
31 32 33
c c c
C c c c
c c c
=
.
CT (i, j) cji = (1)i+j mji = (1)
i+jij,
11 21 31
12 22 32
13 23 33
T
c c c
C c c c
c c c
=
. (2.4.1)
( ).
( ), CT= A Iv.
.
, C = [cij] cij = (1)i+j mij =
(1)i+jij ij i = 1, 2,..., j = 1, 2,..., CT C, . :
) A CT = A Iv.) A 0.
1 1 TA CA
= (2.4.2)
D11 D3 : AA1= A A1= I= 1.
:
,
1 1 .AA
= (2.4.3)
-
68
2.4.1.
3 1 02 4 10 1 3
A
=
.
. ,
3 1 02 4 1 3 12 1 1 6 0 27 00 1 3
( ) ( )A = = = .
mij = ij ij
1 1( ) ( )i j i jij ij ijc m M+ += =
i = 1, 2, 3 j = 1, 2, 3
1 1 1 2 1 311 12 13
4 1 2 1 2 41 12 1 11 1 6 0 6 1 2 0 2
1 3 0 3 0 1( ) , ( ) ( ) , ( ) ,c c c
2 1 2 2 2 321 22 23
1 0 3 0 3 11 3 0 3 1 9 0 9 1 3 0 3
4 3 0 3 0 1( ) ( ) , ( ) , ( ) ( ) ,c c c+ + += = = = = = = =
=
3 1 3 2 3 331 32 33
1 0 3 0 3 11 1 0 1 1 3 0 3 1 12 2 10
4 1 2 1 2 4( ) , ( ) ( ) , ( ) ,c c c+ + += = = = = = = = =
, =2
=
.
(), A= 0. , (2.4.2)
C
=
, T C
= .
1 1 1 .T
A C
A A = =
(1.4.2), 1 .
-
69
11 6 23 9 31 3 10
C
=
,
11 6 2 11 3 13 9 3 6 9 31 3 10 2 3 10
T
TC
= =
.
, (2.4.2)
111 3 1 11 27 3 27 1 27
1 1 6 9 3 6 27 9 27 3 2727
2 3 10 2 27 3 27 10 27
/ / /
/ / /
/ / /
TA CA
= = =
.
2.4.2.
11
22
33
0 00 00 0
a
A a
a
=
,
3, ( ).
. D2 , -
, A = 11 22 33. 11, 22, 33 , A= 11 22 33 =
0, . 11 22 33 0, (. 11, 22, 33 ) A= 11 22 33 0 (2.4.2).
11, 22, 33
22 11 1111 22 33
33 33 22
0 0 00 0 0
, ,a a a
m m ma a a
= = =
D2
11 22 33 22 11 33 33 11 22, ,m a a m a a m a a= = = .
( ), C=[cij]
22 33
11 33
11 22
0 00 00 0
a a
C a a
a a
=
.
-
70
, , (2.4.2)
22 331
11 3311 22 33
11 22
0 01 1 1 0 0
0 0
T
a a
A C C a aA A a a a
a a
= = =
11
122
33
1 0 00 1 00 0 1
/
/ .
/
a
A a
a
=
- . , 11 22 ... 0,
11
22
0 00 0
0 0 0
...
...
... ... ... ...
aa
A
a
=
11
221
1 0 00 1 0
0 0 0 1
/ ...
/ ...
... ... ... ...
/
aa
A
a
=
. (2.4.4)
11= 22 = 33 =...= a=1
1 0 0 00 1 0 00 0 1 0
00 0 0 1
...
...
...
... ... ... ...
...
I
=
1 I I = .
2.4.3.
, A= 2, = 4, .
1 1 1 1 3 1 2 1 1 2, , ( ) , ( ) , ( ) .T T T TA B AB A B A A B
A B A B AB
. (2.4.4) D10, D11 (. 2.3),
-
71
1 1
1 1
1 1 1 1
223 1 2 1 3 1 2 1
3 3
21 2 1 2 2
1 1 4 22
1 1 124 2
1 1 1 1 1 1 116
1 1 12
1
,
,
( ) ( )
( ) ( )
( ) ( )
T TT T
T T
T T TT T T T
A B A B BA
AB A B AB
A B A A B AA A A B AA B A B
B AA B A B A B A B B A
BA A B
A B AA B AB A B AB A B
A B A B
= = = =
= = = =
= = = = =
= = = =
= = = =
2
4.B
BA B
= =
.
2.4.1. . , ,
2 2 6 2 1 1 1 1 1 1 1 11 2 1 1 1 2 1 1 2 2 3 31 1 3 1 1 1 3 1 2
2 2 21 1 1 4 1 1 1 4 1 2 3 4
, ,
.
2.4.2.
1 2 4 8 1 1 1 1 1 1 1 12 4 1
1 3 9 27 1 0 1 1 1 2 1 11 1 1
1 1 1 1 1 1 0 1 1 1 2 13 9 1
1 4 16 64 1 1 1 1 1 1 1 1
, , , .
2.4.3. , , A=== 2, .
1 1 1 3 1 1 1 1 2 1 1, ( ) , , ,T T TA B A B A B A B B A .
2.4.4. , 1 = 2.
2.4.5. , 1.
2.4.6. x , -.
1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1
, ,
x x x
x x
x x x
.
-
72
2.5 .
2, , , - .. . .
. 2.5 Ohm (),
mper () volt (V).
+
+
_
_
8
5 10 38 V
40 V
I1 I3
I2
. 2.5.
Kirchhoff - 1, 2, 3,
1 2 3
1 2
2 3
0
8 10 38
10 5 40
I I I
I I
I I
+ =
+ =
+ =
(2.5.1)
. 4 , , , .
1, 2, 3, - , . 1 20 , 40 , 120 , 20 - . 2 40 , 60 , 40 - 60 . ,
3 60 , 60 , 40 , 40 . 400 , 500 , 520 380 , 1, 2, 3, - .
II, x1, x2, x3 1, 2, 3 , .
x1 1 20 x1 , x2 2 40 x2 x3 3 60 x3 - . , :
20 x1 + 40 x2 + 60 x3
-
73
400 , (- )
20 x1 + 40 x2 + 60 x3 = 400. (2.5.2)
, 500, 520, 380 , , -,
1 2 3
1 2 3
1 2 3
40 60 60 500
120 40 40 520
20 60 40 380
.
x x x
x x x
x x x
+ + =
+ + =
+ + =
(2.5.3)
(2.5.2), (2.5.3) .
, a1x1 + a2 x2 + a3 x3 +...+ a x = , a1, a2, a3,..., a, x1, x2,
x3,..., x, , . - (1, 2, 3,..., ), .
:
x1 = 1, x2 = 2, x3 = 3,..., x = .
, (2, 3, 4) 20 x1 + 40 x2 + 60x3 = 400, 20 2 + 40 3 + 60 4 =
400.
, x1, x2, x3, (2.5.2) (2.5.3). x1, x2, x3 4 -
1 2 3
1 2 3
1 2 3
1 2 3
20 40 60 400
40 60 60 500
120 40 40 520
20 60 40 380
.
x x x
x x x
x x x
x x x
+ + =
+ + =
+ + =
+ + =
(2.5.4)
- 43. , (2.5.1) 33 I1, I2, I3.
, , - :
11 1 12 2 1 1
21 1 22 2 2 2
1 1 2 2
...
...
..................................
...
x x x
x x x
x x x
+ + + =
+ + + =
+ + + = .
(2.5.5)
-
74
ij ,i = 1,2,3,..., j = 1, 2, 3,... , 1, 2,..., . aii (. 11, 22,
33 ...) . - - .
-, . (2.5.1)
1 2 3
1 2 3
1 2 3
0
8 10 0 38
0 10 5 40
I I I
I I I
I I I
+ =
+ + =
+ + =
(1,3,2). , -
. , .
. -, :
3 1
3 2 7
x y
x y
=
+ = ( )
2 4
1
3 =5
x y
x y
x y
+ =
+ + =
( )
, (1,2). ( )( ).
, . , . - :
()
()
( )
1, 2,..., ,
11 1 12 2 1
21 1 22 2 2
1 1 2 2
0
0
0
...
...
.........................
...
x x x
x x x
x x x
+ + + =
+ + + =
+ + + =
-
75
. - (0,0,0,,0) , -
. , ( ), , .
. , , (2.5.4)
1 2 3
1 2 3
1 2 3
1 2 3
20 40 60 400
40 60 60 500
120 40 40 520
38020 60 40
.
x x x
x x x
x x x
x x x
+ +
+ + =
+ + + +
(2.5.6)
,
20 40 6040 60 60120 40 4020 60 40
A
=
, 1
2
3
x
X x
x
=
,
400
500
520
380
B
=
41 (2.5.6) ,
1 2 3
11 2 3
21 2 3
3
1 2 3
20 40 6020 40 6040 60 6040 60 60
120 40 40 120 40 4020 60 40 20 60 40
x x xx
x x xAX x
x x xx
x x x
+ + + + = = + + + +
(2.5.6) 1
2
3
40020 40 6050040 60 60
120 40 40 52020 60 40 380
x
x
x
=
= . , (2.5.1) =
1
2
3
1 1 1 08 10 0 380 10 5 40
, ,
I
A X I B
I
= = =
.
, (2.5.1)
-
76
11 12 1 11
21 22 2 22
1 2
...
...
......
...
x x
x
=
. (2.5.7)
=, 11 12 1
21 22 2
1 2
...
...
...
A
=
1
2
...
x
xX
x
=
,
1
2
...
B
=
() () . -
, , . . -
, , - Gauss, .
2.7 , Cramer, - , , (. ).
, - .
- . . , . ( ) :
) . , , .
) . - . , ( ). , .
-
77
) . - . ( ).
, - (), , , .
(2.5.1), , - 1 3
2 21 3
38 10 40 108 5
,I I
I I
= =
'
2 22
38 10 40 10 08 5
I II
+ = .
2=3
1 338 10 3 40 10 31 2
8 5,I I
= = = = .
1=1, 2=3, 3=2.
2.5.1.
1 2 3
1 2 3
1 2 3
2 3
1
3 2 7
x x x
x x x
x x x
+ =
=
+ = .. x1 (
, ),
x1 = 1 + x2 + x3
-
2 3 2 3
2 3 2 3
2 1 3
1 3 2 7
( )
( )
x x x x
x x x x
+ + + =
+ + + =
2 3
2 3
3 5
2 3 8.
x x
x x
+ =
+ =
x2 x2 = 5 3x3 -
2(5 3x3) + 3x3 = 8.
9x3 = 18 x3 = 2,
-
78
x2 = 5 3x3 = 5 3 2 = 1,x1 = 1 + x2 + x3 = 1 +(1) + 2=0.
(x1, x2, x3) = (0, 1,2).
2.5.2.
1 2 4
1 2 3
2 3 4
3
2 2
3 3
x x x
x x x
x x x
+ =
+ =
+ =. x1, x1 = x2 x4 + 3 -
( x1)
2 4 2 3
2 3 4
2 3 2
3 3
( )
x x x x
x x x
+ + =
+ =
2 3 4
2 3 4
3 2 4
3 3.
x x x
x x x
=
+ =
, x3, x3 = 3x2 2x4 + 4 3x2 + (3x2 2x4 + 4) x4 = 3 6x2 3x4 =
1.
, - . , x4 = R
42
3
1 2 4
3 1 3 16 63 1 21 3 73 2 46 6 2
3 1 17 33 36 6
,
,
.
x x
x
x x x
= =
= + = =
= + = + =
1 2 3 417 3 3 1 7
6 6 2( , , , ) , , , ,
x x x x
=
R .
2.5.3.
1 2 3
1 2 3
1 2 3
1 2 3
6
2 5
5 2 1
2 10
x x x
x x x
x x x
x x x
+ + =
+ =
+ =
+ =
-
79
. x1, x1 = x2 x3 + 6
2 3 2 3
2 3 2 3
2 3 2 3
6 2 5
6 5 2 1
6 2 10
( )
( )
( )
x x x x
x x x x
x x x x
+ + =
+ + =
+ + =
2 3
2 3
2 3
2 1
4 3 5
2 4.
x x
x x
x x
+ =
=
=
, x3, x3 = 2x2 1
2 2
2 2
4 3 2 1 5
2 2 1 4
( )
( )
x x
x x
=
=
2
2
2 8
3 2.
x
x
=
=
x2 = 4, . .
120 cm10 cm
20 cm
20 cm
30 cm50 cm
A
A
B
B
. 2.5.
120 cm10 cm
20 cm
20 cm
30 cm50 cm
A
A
B
B
. 2.5.
.
2.5.1. 1000 kg, , , , . . . , , - 2.5 2.5.) x, y, z ,
x, y, z.) , -
.
-
80
2.5.2. (. 2.5) (1,0) y = ax2 + x + . - (x, y)=(2, 8), (5,
20).
2.5.3. (2.5.5) - .
1 11
2 22
3 3
1 1 1 0 1 1 02 1 1 3
1 1 1 0 1 2 01 1 1 1
2 1 1 0 3 2 01 3 2 7
3 1 1 0 2 2 0
, ,
x xx
x xx
x x
= = =
1 11
2 22
3 33
4 4
1 1 1 2 2 1 1 1 22 1 1 1 1 2 1 1 1 23 1 1 0 4 3 1 1 0
,
x xx
x xx
x xx
x x
= =
.
2.5.4. AX=B, , .
) 1 2 3
1 2 3
1 2 3
2 3 9
3 5 6 1
2 8 3
x x x
x x x
x x x
) 5
2
3 3 12
x y z
x y z
x y z
+ + =
+ =
+ + =
) 3 0
0
3 2 0
6
x
x
x
x y z
=
+ =
=
+ =
) 2 3 7
4 9 3
4 3 0
7 3 4
x y z
x y z
x y z
x y z
+ + =
+ + =
+ =
+ + =
) 1 2 3
1 2 3
1 2 3
1 2 3
5 2 3 0
2 3 3 2
2 5 2 3
2 3 2 1
x x x
x x x
x x x
x x x
+ + =
+ + =
+ + =
+ + =
) 2 3 4 5 6
1
2
3
4
x y z
x y
y
y z
x y z
+ + + =
+ =
+ =
+ =
+ + =2.5.5. .
) 5
2
3 3 12
x y z
x y z
x y z
+ + =
+ =
+ + =
) 10
3
5
6
x y z
x y
y
y z
) 2 7
3 9 13
0
4
x y z
x y z
x y z
x y z
+ + =
+ + =
+ =
+ + =
) 3 6
2 4
2 4
2 4
3
x y z
x y z
x y z
x y z
x y z
+ + =
+ + =
+ + =
+ + =
+ + =
20
15
10
5
42 6 8 10O
. 2.5.
-
81
2.5.6. 120 m. 40 m - , .) x, y, z ,
x, y, z.) , -
.
2.5.7. , , , . , . 2 , 1 3 . 3 2 , , - 4 4 . , 90 , 30 70 - , .)
x, y, z , , ,
x, y, z.) , , , -
.
2.5.8. II , - . 1 , 4 - 3 , 2 , 3 1 . 15 , 35 20 ,) x, y
II , , .
) x, y .
2.6 .
, . , - , ' . , - ....
' , , .
, - .
, ' :
-
82
1. i j ( i j )
i j
2. i ( i) ( 0)
i i
3. i ( i) j ' ( 0).
i i+j
, ( ).
, , -, . , , . , .
(2.5.4) - II 2.5.
i i, 20, ( , ). ,
1 1 2 3 420
, , , ,i i i =
:
1 2 3
1 2 3
1 2 3
1 2 3
2 3 20
2 3 3 25
6 2 2 26
3 2 19 .
x x x
x x x
x x x
x x x
+ + =
+ + =
+ + =
+ + =
(1)
, 1 (1) 2 2 (1). 2
x1 . ,
2 2 + (2) 1 :
1 2 3
2 3
1 2 3
1 2 3
2 3 20
3 15
6 2 2 26
3 2 19
.
x x x
x x
x x x
x x x
+ + =
=
+ + =
+ + =
(2)
, 1 (2) 6
-
83
3 [ 3 3 + (6) 1]. 3 x1 :
1 2 3
2 3
2 3
1 2 3
2 3 20
3 15
10 16 94
3 2 19
.
x x x
x x
x x
x x x
+ + =
=
=
+ + =
(3)
1 (3) 1 4 [ 4 4 + (1) 1]. x1 4 , :
1 2 3
2 3
2 3
2 3
2 3 20
3 15
10 16 94
1
.
x x x
x x
x x
x x
+ + =
=
=
=
(4)
(4) x1 - . x2 .
3 3 + (10) 2 :
1 2 3
2 3
3
2 3
2 3 20
3 15
14 56
1
.
x x x
x x
x =
x x =
+ + =
=
(5)
4 4 + 1 2 :
1 2 3
2 3
3
3
2 3 20
3 15
14 56
4 16
x x x
x x
x =
x =
+ + =
=
(6)
' , x1 x2 .
45 4 314( ) + , :
1 2 3
2 3
3
3
2 3 20
3 15
14 56
0 0
.
x x x
x x
x =
x =
+ + =
=
(7)
-
84
, ( ), - (, : 0x1 + 0x2 + ... + 0x = 0, ).
, (7) ,
1 2 3
2 3
3
2 3 20
3 15
14 56
.
x x x
x x
x =
+ + =
=
(8)
x3 = 4 x3 = 4 (8), x2 = 3. x2 = 3 x3 = 4 (8), x1 = 2.
(8) (2.5.3), - (2.5.3) (2,3,4),
x1 = 2, x2 = 3 x3 = 4.
, , :
1: , i j, , x1 . - - .
2: x1 , - i i + j .
3: , 1 2 1 1 . 2, 3, 4, , 2 , -.
, , : ) N , ( -
, i i),
) N - 1 ( - i j i i).
, - (2.5.3) ( (4) ) :
2 (1)2 (4), x2 1, :
1 2 3
2 3
2 3
2 3
2 3 20
3 15
10 16 94
1
.
x x x
x x =
x x =
x x =
+ + =
+
(5)
-
85
' 3 3 +10 ^ 2 :
1 2 3
2 3
3
2 3
2 3 20
3 15
14 56
1
x x x
x x =
x =
x x =
+ + =
+
(6)
, 4 4 + (1) 2, :
1 2 3
2 3
3
3
2 3 20
3 15
14 56
4 16
.
x x x
x x =
x =
x =
+ + =
+
(7)
x3 1 3 13 314 ( ) 3, :
1 2 3
2 3
3
3
2 3 20
3 15
4
4 16
.
x x x
x x =
x =
x =
+ + =
+
(8)
, 4 4 + 43 :
1 2 3
2 3
3
3
2 3 20
3 15
4
0 0.
x x x
x x =
x =
x =
+ + =
+
(9)
.
2.6.1.
1 2 4
1 2 3
2 3 4
3
2 2
3 3
x x x
x x x
x x x
( ,
2.5.1 ).
.
2 2 + (2) 1, 1 2 4
2 3 4
2 3 4
3
3 2 4
3 3.
x x x
x x x
x x x
-
86
x2 1, 2 213
,
1 2 4
2 3 4
2 3 4
3
1 2 43 3 3
3 3
.
x x x
x x x
x x x
+ = = + =
3 3 32
1 2 4
2 3 4
3 4
3
1 2 43 3 32 7
x x x
x x x
x x
+ = = + =
.
x4 = , R 3 372 72
x x
= = .
H 21 7 2 4 7 4 8 3 13 2 3 3 6 6
x
+ = + = =
.
, x2, x3
1 2 43 1 18 3 1 6 17 33 36 6 6
x x x +
= + = + = = .
1 2 3 417 3 3 1 7
6 6 2( , , , ) , , , ,
x x x x
=
R .
2.6.2.
1 2 3 4
1 2 3 4
1 2 3 4
2 3 4
3 2 1
3 8 2 5 11
6 5 2 4 4
4 2 5
x x x x
x x x x
x x x x
x x x
+ + =
+ + =
+ =
+ =
.
. 2 2 31
1 2 3 4
2 3 4
1 2 3 4
2 3 4
3 2 1
4 2 8
6 5 2 4 4
4 2 5.
x x x x
x x x
x x x x
x x x
+ + =
+ =
+ =
+ =
-
87
3 3 +(6)1
1 2 3 4
2 3 4
2 3 4
2 3 4
3 2 1
4 2 8
13 10 10 10
4 2 5.
x x x x
x x x
x x x
x x x
+ + =
+ =
=
+ =
x2 1, 3 3 +( 13)2
1 2 3 4
2 3 4
3 4
2 3 4
3 2 1
4 2 8
42 36 114
4 2 5.
x x x x
x x x
x x
x x x
+ + =
+ =
=
+ =
4 4 + (1)2
1 2 3 4
2 3 4
3 4
3 4
3 2 1
4 2 8
42 36 114
0 0 3
x x x x
x x x
x x
x x
+ + =
+ =
=
+ =
x3, x4, .,
0x1 + 0x2 + ... + 0x = 0, . - .
.
2.6.1. .
)1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
2 5 6 12 3 2 2
2 4 2 6 3 3
x x x x x
x x x x x
x x x x x
+ + + = + = + + + =
)
1 2 3
1 2 3
1 2 3
2 3 5
3
3 2 7
x x x
x x x
x x x
+ =
+ =
+ =
)
2 0
2 5 3
4 5 7 3
3 2 1
x y z
x y z
x y z
x y z
+ =
+ =
+ =
+ + =
)
1 2 3
1 2 3
1 2 3
1 2 3
1 2 3
6
2 5
5 1
2 3 3
4 3 2
x x x
x x x
x x x
x x x
x x x
+ + =
+ =
+ =
+ + = + =
-
88
2.6.2. .
)
2 5
8
4
y z
x z
x y
+ =
+ = + =
)
1 2 4
1 2 3
2 3 4
3
2 2
3 3
x x x
x x x
x x x
+ =
+ =
+ =
)
3 4
3 4
1 2 3 4
2 3 4
1
6 5 2
3 2 12
4 2 4
x x
x x
x x x x
x x x
=
+ =
+ + =
+ =
)
2 3 4
1 2 3 4
2 3 4
1 2 3 4
3 4
4 2 8
3 2 1
8 2 5 11
6 5 2 4 4
8 17
x x x
x x x x
x x x
x x x x
x x
+ =
+ + =
+ + =
+ = =
2.7 .
, - . , (. 1.4.1)
= = 1 . (2.7.1)
( ).
11 1 12 2 1 1
21 1 22 2 2 2
1 1 2 2
...
...
.....................................
... .
x x x
x x x
x x x
+ + + =
+ + + =
+ + + =
()
2.5, () (. 2.5.5 =)
11 12 1 1 1
21 22 2 2 2
1 2
...
...
...
x x
x
# # % # # # .
= ,
11 12 1
21 22 2
1 2
...
...
...
A
=
,
1
2
...
x
xX
x
=
,
1
2
...
B
=
, () () .
() - . ,
-
89
(2.7.1),
= 1 .
A 0. (2.4.2).
, . - , . , .
2.7.1.
)
1 2
1 2 3
2 3
3 1
2 4 1
3 1
x x
x x x
x x
+ =
+ + =
+ =
)
1 2
1 2 3
2 3
3 0
2 4 0
3 1
x x
x x x
x x
+ =
+ + =
+ =
)
1 2
1 2 3
2 3
3 1
2 4 0
3 1
.
x x
x x x
x x
+ =
+ + =
+ =
.) =,
3 1 02 4 10 1 3
A
=
,
1
1
1
B
=
.
, 2.4.1,
111 27 3 27 1 27
1 6 27 9 27 3 272 27 3 27 10 27
/ / // / // / /
TA CA
1
1 1 311 27 3 27 1 276 27 9 27 3 27 1 02 27 3 27 10 27 1 1 3
// / /
/ / /
/ / / /
X A B = = =
.
x1 = 1/3, x2 = 0, x3 = 1/3.) = 1,
1
001
B
=
.
-
90
11
11 27 3 27 1 27 0 1 276 27 9 27 3 27 0 3 372 27 3 27 10 27 1 10
27
/ / / /
/ / / /
/ / / /
X A B
= = =
.
) AX = B2, -
2
100
B
=
12
1 11 2711 27 3 27 1 276 27 9 27 3 27 0 6 272 27 3 27 10 27 0 2
27
// / /
/ / / /
/ / / /
X A B = = =
A x1, x2, x3
11 1 12 2 13 3 1
21 1 22 2 23 3 2
31 1 32 2 33 3 3
x x x
x x a x
x x a x
+ + =
+ + =
+ + =
= 1 ,
11 12 13
21 22 23
31 32 33
a a a
A a a a
a a a
=
. (2.4.2), ,
111 21 311
12 22 32 2
13 23 33 3
1 1Tc c c
X A B C B c c c A A
c c c
= = =
11 1 21 2 31 31 111 21 31
2 12 22 32 2 12 1 22 2 32 3
13 23 333 3 13 1 23 2 33 3
1 1c c c x c c c
x c c c c c c A A
c c cx c c c
+ + = = + + + +
,
1 1( ) ( )i j i jij ij ijc m M+ += = ij i = 1, 2, 3
j = 1, 2, 3. ,
11 1 21 2 31 3 12 1 22 2 32 3 13 1 23 2 33 31 2 3, ,
c c c c c c c c c x x x
A A A
.
-
91
x1
22 23 12 13 12 131 1 2 1 3 111 22 21 21 31 31
32 33 32 33 22 231 1 1( ) , ( ) , ( )
c M c M c M
+ + += = = = = =
22 23 12 13 12 13
1 2 332 33 32 33 22 23
1 .
x
A
+=
1 12 13
2 22 23
3 32 33
1 ,
1
1 12 13
2 22 23
3 32 331
11 12 13
21 22 23
31 32 33
x
D x
D
.
D
Dx1
1 12 13
1 2 22 23
3 32 33
x
a a
D a a
a a
=
D x1 . - x2, x3,
322 3,
x xDDx x
D D
Dx2 2 3
11 1 13 11 12 1
21 2 23 21 22 2
31 3 33 31 32 3
, ,x x
D D
Dx3 2 3
11 1 13 11 12 1
21 2 23 21 22 2
31 3 33 31 32 3
, ,x x
D D
D x2 x3 .
- (
-
92
1. C. MacLaurin 1729. , Cramer, , 5 , ' 5 5 . (1750).
). , - . ' Cramer1.
Cramer.
= .) 0, (x1, x2, ..., x)
1 21 2, ,...,
x x x
D D Dx x x
D D D= = = ,
(2.7.2)
D Dxi , i = 1, 2, 3, ..., D i - xi .) = 0, .
= 0, = Gauss.
, 2.1, Cramer 22 (. 2.1.8).
Cramer, , ' , . , - , Gauss Cramer. Cramer .
2.7.2.
)
2 3 50
4 5 6 11
x y z
x y z
x y z
+ + = = + + =
)
2 3 10
4 5 6 0
x y z
x y z
x y z
+ + = = + + =
.
)
1 2 31 1 1 6 04 5 6
D = =
-
93
5 2 3 1 5 3 1 2 50 1 1 6 1 0 1 6 1 1 0 1211 5 6 4 11 6 4 5
11
, ,x y zD D D= = = = = = .
Cramer,
2 3 2( , , ) , , ( , , )yx zDD D
x y zD D D
= =
.
(2, 3, 2). ) , D = 6 0.
, .
1 2 3 1 1 3 1 2 10 1 1 1 1 0 1 10 1 1 0 90 5 6 4 0 6 4 5 0
, ,x y zD D D= = = = = =
Cramer,
1 10 96 6 6
( , , ) , , ( , , ).yx zDD D
x y zD D D
= =
2.7.3.
2
2
2
2
2
2.
x y z
x y z
x y z
+ = + = + + =
.
2
2 2 4 2 2 2 2 2 2 2
2
1 1
1 1 1 1 1 1 1 1 1 1 1 1
1 1
( ) ( ) ( ) ( )( ) ( )( )( )
D
2 2 2 20 1 1 0 1 1 0 0 1 1( )( ) ( )( ) .D
2 4 2 2 2 2
2
2
2 2 2 2 2
2
2
2 2 2 2 2 2
2 1 1
2 1 2 1 2 1 2 1 2 1 1
2 1
2 11 2 1 2 1 2 1 2 1 1 2 1 1
1 2
1 2
1 2 2 1 2 1 1 2 1 2 1 11 1 2
( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( )( ).
x
y
z
D
D
D
2 4 2 2 2 2
2
2
2 2 2 2 2
2
2
2 2 2 2 2 2
2 1 1
2 1 2 1 2 1 2 1 2 1 1
2 1
2 11 2 1 2 1 2 1 2 1 1 2 1 1
1 2
1 2
1 2 2 1 2 1 1 2 1 2 1 11 1 2
( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( )( ).
x
y
z
D
D
D
-
94
2 4 2 2 2 2
2
2
2 2 2 2 2
2
2
2 2 2 2 2 2
2 1 1
2 1 2 1 2 1 2 1 2 1 1
2 1
2 11 2 1 2 1 2 1 2 1 1 2 1 1
1 2
1 2
1 2 2 1 2 1 1 2 1 2 1 11 1 2
( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( )( ).
x
y
z
D
D
D
D 0,1,1, :
) 0 1 1 D 0
2 2
2 2 2 22 1 1 2
1 1( )( )
( )( )xD x
D
+= = =
+,
2 2
2 2 2 22 1 1 2
1 1( )( )
( )( )
yD yD
+= = =
+,
2 2
2 2 2 22 1 1 2
1 1( )( )
( )( )zD z
D +
= = = +
.
2 2 21 1 1, ,
.
) = 0, 222
y z
x z
x y
= = + =
.
224,
y z
x z
y z
= = =
.
) = 1,
22
22
2
x y zx y z
x y zx y z
![1001-Solved-Problems-In-Engineering-Mathematics [Appendix C-H. (Physical Constants, Power of 10, Numeration, Math Notation, Greek Alphabets & Divisibility Rule)]](https://img.pdfslide.net/doc/110x75/577c77801a28abe0548c5b76/1001-solved-problems-in-engineering-mathematics-appendix-c-h-physical-constants.jpg)
