Высшая математика. Том 1

+ Maxima,

5 determinant.

/

- , , ,

, $ .

2.6.! " ! " "

1.0$ AT A,

|AT| = |A|, , $

, , -

:

a11	a12	a13	=	a11	a21	a31	.
a21	a22	a23		a12	a22	a32
a31	a32	a33		a13	a23	a33

2.& 2- , -

, , :

a11	a12	a13	= −	a13	a12	a11	.
a21	a22	a23		a23	a22	a21
a31	a32	a33		a33	a32	a31

3. 0$ ,

:

a11	a12	a13	= 0 .
a21	a22	a23
a21	a22	a23

!, $ 2- 3- , 2

, -

, |A| = –|A| |A| = 0.

4. + -

( , k

):

ka11	a12	a13	= k	a11	a12	a13	.
ka21	a22	a23		a21	a22	a23
ka31	a32	a33		a31	a32	a33

61

5.0$ - ,

.

6.0$ - -

2- ,

2- 5:

a11

a12

a13

a11

a12

a13

a11

a12

a13

a21

a22

a23

=

a21

a22

a23

+

a21

a22

a23

.

a′

+ a′′

a′

+ a′′

a′

+ a′′

a′

a′

a′

a′′

a′′

a′′

31

31

32

32

33

33

31

32

33

31

32

33

7. 0$ - ( )

( ), ,

:

a11	a12	a13		=		a11	+ ka12	a12	a13		.
a21	a22	a23				a21	+ ka22	a22	a23
a31	a32	a33				a31	+ ka32	a32	a33

8. 9

, $ :

a11	0	0	=	a11	a12	a13	= a11	× a22	× a33 .
0	a22	0		0	a22	a23
0	0	a33		0	0	a33

2.7.& - * ) ".

" " " ! " "

:

∆ =	a11	a12	a13	.
	a21	a22	a23
	a31	a32	a33

, $ aij -

, , -

, ,

i- j- . ) aij -

Mij.

, M12, $ a12, -

: M12	=	a21	a23	, 1-
		a31	a33

2- .

0$ 5

a11a22a33 + a12a23a31 + a21a13a32 − a13a31a22 − a12a21a33 − a32a23a11

62

a11 (a22a33 − a32a23 ) − a12 (a21a33 − a23a31 ) + a13 (a21a32 − a31a22 ) ,

, $ 1-

; , $ a12,

«–», , $ :

∆ = a11M11 − a12M12 + a13M13

(2.1)

! :

∆ =

a11

a12

a13

= a

a22

a23

− a

a21

a23

+ a

a21

a22

.

a

a

a

21

22

23

11

a

a

12

a

a

13

a

a

a31

a32

a33

32

33

31

33

31

32

. , 5

a11, a12, a13 -

.

; ,' . . .

& 1. - :

,		2	3		−4				= 2				0 −2					− 3						1		−2					− 4		1 0				= 8 − 3 (−1) − 4 2 = 3 ./
		1 0			−2
		−1	2		1								2			1							−1		1								−1		2
& 2. - :
,		1	0		−2			= 1			1 1					−		2		−1				1		= 1 + 2							4 = 9./
		−1	1	1
		3	1		2						1			2						3				1
& 3. 1’																												x + 3					2 + x				4		= 0 .
																													3				x − 1				3
																													4				x				4
,( x + 3)					x − 1 3								− (2			+ x)						3 3				+ 4					3 x − 1				= 0 .
					x		4															4		4							4		x

(x+3)(4x–4–3x)+4(3x–4x+4)=0. (x+3)(x–4)+4(–x+4)=0.

(x–4)(x–1)=0. x1 = 4, x2 = 1./

" $ .

aij -

Mij, (–1)i+j.

; aij Aij.

+ , $ '

63

Aij = (–1)i+j·Mij, -

– .

!

:

+

−

+

− ...

−

+

−

+ ...

+

−

+

− ...

−

+

−

+ ...

... ... ... ... ...

1. 2.8. &

,

A = (−1)1+1

M

11

= M

11

;

A = (−1)3+2 M

32

= −M

32

.

11

32

!

0

−1

1

. + A13, A21, A32.

3

2

−2

4

−1

2

, A =

2 3

= −2

− 12 = −14 ;

A = −

−1 1

= 2 − 1 = 1;

A = −

0 1

= 2 ./

13

4

−1

21

−1

2

32

2 −2

4 , $ , 5 (2.1) :

∆ = a11 (−1)1+1 M11 + a12 (−1)1+2 M12 + a13 (−1)1+3 M13 = a11A11 + a12 A12 + a13 A13 .

; 5

- .

, 2- -. 9 ,

, :

∆ =

a11

a12

a13

= −

a21

a22

a23

.

a21

a22

a23

a11

a12

a13

a31

a32

a33

a31

a32

a33

1 1- .

a21

a22

a23

= a

a12

a13

− a

a11

a13

+ a

a11

a12

(2.2)

a

a

a

22

23

11

12

13

21

a

a

a

a

a

a

a31

a32

a33

32

33

31

33

31

32

+ ∆ =

a11

a12

a13

= −a21M 21 + a22M22 − a23M23 ,

a21

a22

a23

a31

a32

a33

$ 5 (2.2) a21,

64

a22, a23.

. , ∆ = a21A21 + a22 A22 + a23 A23 , 5

2- .

; 5

. 9

( ), , $

.

. , :

" .+ !$ )$" " )$ - * - *

() ! ") * ) ".

.

			& 1. -									2			1		−2		, -
												−1			0		3
												−3			4		1
2- .
,		2	1	−2		= (−1)1+2 1		−1 3		+ 0 + (−1)3+2 4				2	−2		= −(−1+ 9) − 4(6 − 2) = −24 ./
		−1	0	3
		−3	4	1				−3 1						−1	3

& 2. - 4- ,

.

, 9 , -

, - , ,

. " ,

:

					4	1		1		0					4	1		1	0			4	1	1
				−2 −1 0 1								=			−4 0			−2 0			= 1				.
																						−4 0		−2
				−2		1			−2	−1					−2	1		−2		−1		0	1	−1
					0	1			−1	0					0	1		−1	0
. ,
3- :
		4	1		1		=		4	1		2			= −1		4	2		= − (−8 + 8) = 0 ./
		−4 0			−2				−4 0		−2
		0	1		−1				0	1		0					−4 −2

2.8. "$ " , 2.2

6 ’ ,

.

- - ,

- .

65

	1
1.	∆ = 0
	3
	1
	1
3.	∆ = 2
	3
	2
	1
5.	∆ = −1
	0
	1
	5
7.	∆ = −1
	2
	1
	1
9.	∆ = −1
	0
	1
	2
11.	∆ = 1
	3
	1

2

13.∆ = −1

0

3

1

0

−3

015. ∆ =

2

17. ∆ = 02

1

4		1		3
−1		3	−1
1		0		2 .
−2		5		1
2		3		−2
−1		−2		−3
2		−1		2 .
−3		2		1
−1		−1			1
2		2			0
−1		1			4	.
1		−1		−1.5
−4		0		2
1		1		−1
3		1		−6 .
0		2		−1
2		0			1
−3		3			−1
4		−10			2 .
−1		1			3
−1		1		2
2		−1		1
0		−1		−3 .
−1		1		3
3		0		1
1		3		0
2		−1		1 .
−1		1		−2
0		3		4
1		5		1
4		10		1 .
−6		0		−1
1		0	1
3		2	0
1		2	0 .
0	−2		1

66

	1	1	1	1
2. ∆ =	1	4	2	3
	1	10	3	6 .
	6	10	1	4

−2

4. ∆ = 1

2

−5

3

6. ∆ = 23

1

4

8.∆ = 3

0

−1

1

0

−1

110. ∆ =

	2
12.	∆ = −1
	1
	1
	−1
14.	∆ = −2
	0
	−1
	3
16.	∆ = −2
	0
	−1

−2

18.∆ = 2

3

4

2	1	0
−3	3	7
−1	2	−1 .
4	1	2
−2	2	0
1	2	−2
−1	2	1 .
2	−1	−1
−1	0		1
2	−1		2
2	2		1 .
1	−3 −1
4	−3	0
4	1	2
2	4	1	.
0	−1	5
1	2		0
−3	3		−1
3	−8		1 .
−1	2		−1
3	3	2
2	2	1
1	2	0 .
3	3	3
1	3	2
2	2	1
1	2	0 .
3	3	3
−2	−1		3
1	0		−1
2	1		−3 .
3	2		−4

1

19. ∆ = 32

1

1

21. ∆ = 23

4

1

23. ∆ = 22

1

2

25. ∆ = 13

0

1

27. ∆ = 23

4

2

29. ∆ = 33

2

3

31. ∆ = 22

2

1		2	3
−1	−1		−2
3	−1		−1 .
2		3	−1
2	3	4
1	2	3
2	1	2 .
3	2	1
2		−1	1
−1		1	2
1	−2		−6 .
−1		1	3
1	−3		0
5		1	2
6		4	1 .
−1	−1		5
2	3	4
1	2	−3
2	1	−2 .
3	2	1
1		1	−1
−1		0	−3
0		−1	1 .
2	−2		5
0		−1	1
1		1	−1
−1		0	−3 .
2	−2		5

1

20. ∆ = 03

1

1

22. ∆ = 02

6

2

24. ∆ = 10

1

2

26. ∆ = 33

3

1

28. ∆ = 23

4

2

30. ∆ = 13

4

2.9. /"

4	1		3
−1	3		−1
1	0		2 .
−2	5		1
1	−3		0
5	1		2
6	4		1 .
−1	−1		5
5	−3		2
−4	5		−2
3	−8		1 .
−1	2		−1
−1	3		2
3	3		2
−1	−1		2 .
−1	3		−1
2	3	4
3	4	1
4	1	2 .
1	2	3
1	−3		4
0	−2		3
2	0		−5 .
3	−5		0

& . 0$ A – , -

, $ A–1 :

A A−1 = A−1 A = E .

6 :

! $ A , -

, $ .

67

0$

,

a11

a12

a13

A

=

a22

a23

: A

−1

=

A

,

a21

A

a31

a32

a33

~
A – .		A11		A21	A31
6	A						,
		=	A12	A22	A32
			A	A	A
			13	23	33

Aij – aij A.

!

A,

..

- , $ : 1. + A;

2. + ;

3. + Aij

~

( A);

4.+ 5: A−1 = A .

A

; , -

A	−1	=	1		A	A
					A	A	.
					11	21
			A		12	22

+ :

1)–1

, :

det A–1=1/det A;

2)

, :

( )–1 = –1 –1;

3) -

, :

( –1)T = ( T)–1.
& 1. + , A = (10	−21). +
.
, + : |A| = 2.	0).
+ : AT = ( 1
−1	2

68

+ : A = (2			1).
	0		1						(02	11)= (10 1/1/	22).
+ : A−1 =				A			=	1
				A				2
&: A−1 A = E
(10 1/1/ 22) (10	−21)= (10 10)= E ./

& 2. + ,	1
	A = 3
	0

, + :
																		A			=				2 1						− 2			3 1			= 3 − 12 = −9 .
																									1	2								0	2
+ :
																																		1	3		0
																										AT =								2	2		1 .
																																		0	1
+ :																																					2
		+		2			1			−		2			1				+				2			2
				1			2					0			2								0			1
																																		3			−4	2
						3	0							1	0									1		3
		−							+												−													−6 2 −1
A =						1	2							0	2									0		1						=							.
					3		0						1		0								1			3								3			−1	−4
		+								−								+
					2		1						2		1								2			2

2	0
2	1	.
1	2

+ : A−1 =		A			1		3	−4	2
				= −			−6	2	−1	.
		A	9					−1	−4
						3

+ Maxima,

5 invert. ! , $ ` 1/|A| -

5 detout.

/

69

2.10. "$ " , 2.3

6 ’ ,

.

+

1.	( · )–1,		2 3 − 1						− 1 0 5
			A = 4 5 2 , B = 0									1 3 .
			−	1 0 7					2			− 2 4
2.	( · )–1,		4		5 − 2					2 1 − 1
			A = 3			− 1 0 , B =				0 1 3 .
			4			2		7		5		7	3
3.	( · )–1,				5		1	7		2		4	1
			A = − 10 − 2 1 , B = 3 1 0 .
					0		1	2		7		2	1
4.	( · )–1,			7		2		0		0		2	3
			A = − 7			−	2	1 , B =		1 0			− 2 .
			1			1		1		3		1	1
5.	( · )–1,		5			2 0			3		6 − 1
		A = 10				4 1 , B			= − 1		−	2 0 .
			7			3	2		2		1		3
6.	( · ) –1,		5 − 1 3							3 7 − 2
			A = 0			2		− 1 , B = 1 1					− 2 .
			−		2	− 1 0				0 1 3

7.	( · ) –1,	5 3 − 1			2	4 16
		A = 2 0 4 , B = − 4				− 2 0 .
		3 5		− 1	6	8		2
8.	( · ) –1,	3 2 − 5			− 1	2 4
		A = 4 2		0 , B = 0		3	2 .
		1 1 2			− 1	− 3 4
9.	( · ) –1,	1 4 2			4 6 − 2
		A = 2 1		− 2 , B = 4 10 1 .
		0 1		− 1	2 4		− 5
10.	( · ) –1,	4	2	1	2	0	2
		A = 3	− 2	0 ,	B = 5	− 7	−	2 .
		0	− 1 2		1 0		−	1
11.		1	0	3	7	5	2
	( · ) –1, A = − 2		0	1 ,	B = 0	1	2 .
		− 1 3 1			− 3	− 1	− 1
	( · ) –1,	2	3	1	2	7	13
12.		A = − 1	2	4 ,	B = − 1	0	5 .
		5	3	0	5	13	21

70