Высшая математика. Том 1

13.	( · ) –1,	1 − 2 3								4 11 3
		A = 2				3			5 ,	B = 1			6		1 .
		1 4						− 1		2 2 16
14.	( · ) –1,	2				3		1		9			8	7
		A = 4			− 1			0 ,		B = 2			7	3 .
		0				1		2		4			3	5
15.	( · ) –1,	2			1			3		1			2	3
		A = 1			−	2		0 ,		B = − 3			0	1 .
		4			−	3 0				2 1 2
16.	( · ) –1,	4 − 2 0								0 − 2 4
		A = 1			1			2 ,		B = 1			2		3 .
		3			− 2 0					3 4					− 5
17.	( · ) –1,	1 − 1 0									5 3 1
		A = 2			0			− 1 ,		B =	−	1 2			0 .
		7 3 2								−		3 0 0
18.	( · ) –1,		3 2 − 4									0		1 − 2
		A =	−	1			0		2 , B = −				1	1	2 .
		−		2 1				− 1		3				− 1 0
19.	( · ) –1,	3 2 − 1								0			3 − 1
		A = 0			− 1			2 ,		B = 2			−	1	2 .
		5 7 1								− 3 1 4
	( · ) –1,	4					5 6			0 − 1					1
20.		A = − 1					0		3 ,	B = 1			0		− 2 .
		− 1 2						− 1		3 1					2
21.	( · ) –1,	7 − 3 0								− 4 2 1
		A = 1			− 1			0 ,		B = 1			0	1 .
		2			0			3		3			2	1
22.	( · ) –1,	1				2			3		4		2		1
		A = 0			− 2				3 ,	B = − 1			2		0 .
		1 1 1								2 3					− 1
23.	( · ) –1,	1 − 2						− 2		0 3 5
		A = 1			1			−	2 ,	B = 4			1	0 .
		1			− 1			−	1	1 1 2
24.	(B· ) –1,	1 − 1 2								− 1 0 2
		A = 3			0			− 2 ,		B = 2				1	1 .
		2			− 1 1					− 2 0 1
25.	( · ) –1,	2 1 − 1								2 − 1 0
		A = 1			0			1 ,		B = 0			2		1 .
		3 1					− 2			1 3				− 1
26.	( · ) –1,	1 2							3			2 3 − 1
		A = 4			−		2		1 ,	B = −			2	0	− 1 .
		0 1						− 1		1 0 1

71

27.	( · ) –1,	1 2 − 1					1 2 − 1
		A = 2	3			0 ,	B = 2	− 1	0 .
		0 2			− 1		1 2		1
28.	( · ) –1,		1	2		3	1	0	2
		A = − 1		0		2 ,	B = 2	3	1 .
			1	2			3	1	0
						1
29.	( · ) –1,	2 3 4					2 0 − 2
		A = 1	− 2			0 ,	B = 1	1	0 .
		0 1			2		1	− 1 1
30.	( · ) –1,	2 5			− 1		1 − 2 0
		A = 0		2		1 ,	B = 1	0	2 .
		1		0		1	0	0	3
31.	( · ) –1,	1 2 3					2 − 2 4
		A = − 2		0		5 ,	B = 0	5	− 3 .
		1 2				− 1	1 1		− 1

2.11. 0 ! - * ) " (0)

6 m n :

a	x + a	x + a	x +... +	a x	=b ;
11 1 12 2 13 3				1n n		1
a21x1 + a22 x2 + a23 x3 +... + a2n xn =b2 ;
;
a	x + a	x + a	x +...+ a	x =b		,
	m1 1	m2 2	m3 3	mn n	m

aij bi (i=1, …, m; j=1, …, n) – , x1, …, xn – . " -

5 aij i , j –

, 5.

2 5

		a11	a12	a1n
	A =	a21	a22	a2n		, .
		a	a	a
		m1	m2	mn

#, $ , b1, …, bm

.

6 n C1, …, Cn ’ , $ -

C1, …, Cn x1, …, xn.

’ . & -

:

1. 6 ’ , , {3x1 + 2x2 = 11. 4x1 − 2x2 = 3

1’ ( 1=2; 2=2,5).

72

x1	+ x2	= 0
2. 6 ’ , , {2x	+ 2x	2	= 0 .
1

1’ - , $

.

3. 6 ’ , , {x1 + x2 = 0 . x1 + x2 = 1

0 ’ , x1 + x2 -

.

2.12. " 1 "’ $ 0

1 3- :

a x + a x + a x =b
	11	1	12	2	13		3	1
a21x1 + a22 x2 + a23 x3 =b2 .
a31x1 + a32 x2 + a33 x3 =b3
9 ,			$						,
5 ,				∆ =		a11		a12	a13			-
						a21		a22	a23
						a31		a32	a33

.

6 $ : ∆

,

∆1 =

b1

a12

a13

,

∆2 =

a11

b1

a13

, ∆3 =

a11

a12

b1

.

b2

a22

a23

a21

b2

a23

a21

a22

b2

b3

a32

a33

a31

b3

a33

a31

a32

b3

# (" 1). 0$ < : 0, -

’ , : x1 = ∆∆1 , x2 = ∆∆2 , x3 = ∆∆3 .

0$ , -

’ , ’ , .

& ’ , , -

’ .

x + 2 y − z = 2

& 1. 1’ 2x − 3y + 2z = 2 .

3x + y + z = 8

,∆ =		1	2	−1			= −5+ 2 4 −11= −8 ≠ 0;
		2	−3 2
		3	1	1

		2	2	−1		= −10+ 28− 26 = −8;	∆2	=		1	2	−1		= −14 + 8 − 10 = −16 ;
∆1	=	2	−3 2							2	2	2
		8	1	1						3	8	1

73

		1	2	2
∆3	=	2	−3	2	= −26 − 20 + 22 = −24 .
		3	1	8

x =

∆1

=

−8 = 1, x =

∆2

=

−16 = 2 , x =

∆3

=

−24 = 3 .

∆

1

∆

−8

2

−8

3

∆

−8

- , =1, =2, z=3. /

& 2. 1’

p: {px + 30 y = p + 30, .

30x + py = 0

, 6 ’ , $ < : 0.

∆ =

p

30

= p2

− 302 ≠ 0 . . p ≠ ±30 .

30

p

∆ =

p + 30 30

= p( p + 30) ; ∆ =

p p + 30

= −30( p + 30) .

1

0

p

2

30

0

1.& p ≠ ±30 :

x =

∆1

=

p( p + 30)

=

p

, x =

∆2

=

−30( p + 30)

=

−30

,

1

∆

p2 − 302

p − 30

2

∆

p2 − 302

p − 30

x =

p

, y =

−30

.

p − 30

p − 30

2.

& p=30 30x + 30 y = 60

,

’ .

30x + 30 y = 0

& p=–30 −30x + 30 y = 0 , , -

3.

30x − 30 y = 0

’ x=y, y R. /

& 3. & , -

.

2.4. + -

.

	=			. 2.4
9 -				9
	. ,
				,
	*	*	=
9$	0,2	0,07	0,1	2480
I-	0,1	0,01	0,4	3240
II-	0,05	0,0	0,2	1600

, & x1, x2, x3 ,

. . , , :

74

0,2 x1 + 0,07 x2 + 0,1 x3 = 2480,0,1 x1 + 0,01 x2 + 0,4 x3 = 3240,

0,05 x1 + 0 x2 + 0,2 x3 = 1600.

1’ 2 -

Maxima. ! ’ 64;1 5 2

A, A1, A2, A3

,

. 9 5 2 x	=	∆1	,	x	=	∆2	,	x	=	∆3
1		∆		2		∆		3		∆

` . ! -

5 determinant.

1. 2.9. 7 Maxima

., 8000 , 4000 6000 -

./

75

2.13."’ $" 0 * . "

:

a x + a x + a x =b

11

1

12

2

13

3

1

a21x1 + a22 x2 + a23 x3 =b2 .

a31x1 + a32 x2 + a33 x3 =b3

1

A =

a11

a12

a13

a21

a22

a23

- -

a

a

a

31

32

33

x1

b1

X = x

, B = b

.

x2

b2

3

3

a11

a12

a13

x1

=

a11x1 + a12 x2 + a13 x3

,

+ A X = a21

a22

a23

x2

a21x1 + a22 x2 + a23 x3

a

a

a

x

a x

+ a x

+ a x

31

32

33

3

31 1

32 2

33 3

. 2

:

a11x1 + a12 x2 + a13 x3
a21x1 + a22 x2 + a23 x3
	a	x	+ a	x	+ a	x
	31	1	32	2	33	3

b1

= b , A· X=B.

2

b3

) A B 5, , X

. 3 ’ . -

.

|A| : 0. . -

’ . & -

A–1, A: A−1 ( AX ) = A−1B ,

( A−1 A) X = A−1B .

- A–1A = E E· X = X, ’ -

X = A–1B.

+, $

, ’ -

, . &

’ , , ’ -

.

& 1. 1’

			{3x + 2 y = 7, .
			x − y = 4
, A = (3	2 ), X = (x ),		B = (7). X = A−1B .
1	−1	y	4

76

+ A.
	A		= −5, AT = (3 1 )												, A =						(−1 −2)								, A−1 = −				1	(−1 −2).
								2				−1										−1				3							5 −1		3
	X = A−1B = −							1	(	−1 −2)(7)											= −				1		(−15)= (					3 ).
								5		−1			3				4							5				5			−1
	. , x = 3, y = – 1. /
& 2. 1’ :
A = (2 5), B = ( 5																											XA+B=C,
																	3				), C =							( 5		4 ).
						3	7						−2				−5											−1		−3
, 9 X .
	XA = C − B =							( 5 4 )								− ( 5 3 )												= (	0 1).
								−1				−3						−2						−5					1	2
& A–1 ( -
):											XAA−1 = (C − B) A−1 , X = (C − B) A−1 .
+														–1.
	A		= 2 7 − 3 5 = −1; AT = (52																					73); A = (−73								−25);
	A−1 = − ( 7 −5)= (−7 5 ).
						−3		2					3				−2
	X = (C − B) A−1 =											(	0 1)						(		−7 5 )= (									3 −2).
													1			2					3					−2				−1		1
&: XA = (												3				−2)				(2						5)		= (0		1).
											−1					1					3					7			1	2
	XA + B = (0 1)+ (												5 3 )								= ( 5 4 )									= C ./
							1	2				−2				−5								−1				−3
& 3. 1’
																						x + 2x							= 10,
																								1				2
																						3x1 +						2x2 + x3 = 23,
																						x2				+ 2x3			= 13.
					1 2 0											x1											10
, A = 3 2 1 , X =																x2						, B = 23 .
						0 1 2										x											13
																	3
1’ X = A–1 B.
					1	2	0				2 1							3 1																	1 3 0
	A		=		3 2 1			=						− 2										= 3 − 12 = −9 , AT = 2 2 1 ,
					0 1 2						1 2							0 2																	0 1 2
					3 −4 2										−1							1				3 −4 2
	A =				−6 2				−1			, A					= −									−6 2				−1 .
					3		−1		−4													9				3		−1 −4

77

X = A−1B = −

1

3 −4 2 10

1

−36 4

−6 2

−1

23

= −

−27

= 3 .

9

3

−1

−4

13

9

−45

5

- , 1=4, 2=3, 3=5./

& 4. 1’

A = (

4 ), B = ( 1

AX+B=C,

).

7

4 ), C =

( 2

4

−2

−1

−3

−8

−3

−5

, 8 X = A−1 (C − B).

(C − B)

= ( 2

4 )−

( 1

4 )

= (

1 0).

−3

−5

−3

−8

0

3

A =

−2

+ 8

= 1, A

=

(4

−1), A = ( 2 7 ), A

= ( 2 7 ).

−1

= −7

7 4

T

7

−2

−1

−4

−1

−1

−4

- ,

(−1 −4) (1 0)=

(−1 −12)./

X = A−1 (C − B) =

2

7

0

3

2

21

& 4. "

. + -

, $ -

( 2.5).

	9			. 2.5
				+
&	9 , %			,
				. .
	; 1	; 2	; 3
2003 .	20	22	14	14
2004 .	18	20	18	14
2005 .	21	20	15	14

, + , :

1 2 3

0,18 x1 + 0,20 x2 + 0,18 x3 = 14,

0,21 x1 + 0,20 x2 + 0,15 x3 = 14.0,20 x + 0,22 x + 0,14 x = 14,

1’ -

Maxima. 1’ 5 X = A–1 B, A

B, 5 invert -

.

78

1. 2.10. 7 Maxima

- , , , ,

1=25 . ., 2=25 . ., 3=25 . . /

2.14. "$ " , 2.4

6 ’ ,

.

1’ 64;1:

) 2; ) .

5x1

+ 8x2

− x3

= 12,

x + 2 x

2

+ x

3

= 4,

1.

2.

1

= 1,

x + 2 x

2

+ 3x

3

= 6,

3 x

− 5 x

2

+ 3 x

3

1

= 1.

1

− x3

2 x

− 3x

2

+ 2 x

3

2 x1

+ 7 x2

= 8 .

1

3 x

+ 2 x

2

+ x

3

= 6,

2 x + x

2

+ 4 x

3

= 7,

1

= 6,

1

= 8,

3.

2 x1

+ 3 x2

+ x3

4.

5 x1

+ x2

+ 2 x3

2 x1

+ x2 + 3 x3

= 6.

3 x1

− 3 x2 + x3

= 3.

4 x + 3 x

2

+ 2 x

3

= 3,

2 x − x

2

− x

3

= 0,

1

1

−

= 5,

5.

2 x1

+ 5 x2

+ 3 x3

= 4,

6.

3 x1

+ 4 x2

2 x3

5 x1

+ 6 x2

− 2 x3

= 9.

3 x1

− 2 x2

+ 4 x3

= 5.

x + x

2

+ 2 x

3

= 4,

3 x

− x

2

= 2,

1

= 3,

1

+ x3

= 0,

7.

2 x1

− x2

+ 2 x3

8.

−2 x1 + x2

4 x1

+ x2

+ 4 x3

= 9.

2 x1 − x2 + 4 x3

= 5.

79

3x		− x	+ x =	3,
	1	2	3	= −6,
9. 2x1		− 5x2 − 3x3
x1 + x2 − x3 = 1.

2 x

+ x

2

− x

3

= 2,

1

11.

x1 + x2 + x3

= 3,

3x1 − 6 x2 + x3 = 3.

x

+ 5 x

2

+ x

3

= 7,

1

= 0,

13.

2 x1

− x2

− x3

x1 − 2 x2

− x3

= −2.

3 x + 4 x

2

+ 2 x

3

= 9,

1

− x2

15.

2 x1

− 3 x3

= − 2,

x1 + 5 x2

+ x3

= 7.

2 x

1

+ x

2

+ 4 x

3

= 7 ,

= − 2 ,

17.

2 x1

− x2

− 3 x3

3 x1

+ 4 x2 − 5 x3 = 2 .

x

+ 5 x

2

− x

3

= 5,

1

− x

= 0,

19.

2 x1

− x2

3

3 x1

− 2 x2 +

4 x3 = 5 .

7 x + 5 x

2

+ 2 x

3

= 14,

1

21.

x1

− x2

− x3

= − 1,

x1

+ x2

+ 2 x3 = 4.

x − 2 x

2

− 2 x

3

= −3,

1

+ x2

23.

x1

− 2 x3

= 0,

x1

− x2

− x3 = −1.

x − 2 x

1 + 2 + =

25.2 x1 x2 3x3 6,

2 x1 + 3x2 + 2 x3 = 7.+ x3 = 0,

		2 x + 3x			2	+ 4 x		3	= 9,
		1
27.	3x1		+ 4 x2			− x3 = 6,
	4 x1		+ 5 x2			− 2 x3 = 7.
		3 x	+ x	2	− 2 x		3		= 2,
		1							= 0,
29.		2 x1	− 3 x2			+ x3
	5 x1		+ x2 + 3 x3						= 9.

7x1 + 5x2 + 2x3 = − 15,

31. x1 + x2 + 2x3 = −4,

x1 − x2 − x3 = 1.

1 2 + x3 = 3,

10.2x1 − x2 − 6x3 = −5,

3x1 − 2x2 = 1.x + x

2 x − x

1 + 2 − =

12.3x1 4 x2 5x3 2,

2 x2 + 7 x3 = 9.+ 3x3 = −2,

x − 2 x

2

+ 3x

3

= 2,

14.

1

= 1,

2 x

+ 3x

− 4 x

3x11

− 2 x22

− 5 x33 = −4.

2 x

− x

2

+ 3 x

3

= 4,

1

= 2,

16.

x1 + 3 x2

− 2 x3

2 x2 − x3

= 1.

x − x

2

= 0,

18.

1

+

+ x3

= 6,

2 x1

3 x2

2 x1

+ x2

+ 3 x3

= 6 .

1 1x

+

3 x

2

− x

3

= 1 3,

1

20.

2 x1

+ 5 x2

− 5 x3 = 2,

x1 + x2 +

x3 = 3 .

1 3x2 + x3 = 6,

22.x1 + x3 = 0,

x1 − x2 − x3 = −1.2 x +

1 2 3

24.2 x1 − 3x2 + 4 x3 = 3,

5x1 − x2 + 3x3 = 7.3x + x − 5x = −1,

2x − x = −1,

1 + 2 + =

26.3x1 2x2 x3 6,

2x1 + 3x2 + 3x3 = 8.− 2x3

	2 x − x			2		− 3x	3		= − 2,
		1				+ x3
28.	x1	+ 2 x2					= 4,
	3x1 + x2					− x3	= 3.
	2 x		− x		2	+ 3 x		3	= 4,
		1
30.	x1	+ 3 x2				− x3 = 3,
	5 x1 + 2 x2 + x3 = 8.

80