Высшая математика. Том 1

2.15. - ' *$

" $

, ' , ( ) .

A m×n k, $ -

m n: k ≤ (m,n). 9 k k -

A ( ). 9

, , $

k k , k A.

+ A =	1	2	−1	0
	3	4	−5	6	.
	5	−2	−3	−4

, ) (k=1) - . . 2, –5, –4 – .

) (k=2):

• 1, 2 1, 2 – :

34 = −2 ;

•1, 3 2, 4 – :1 2

2	0	= −8;
−2	−4

• 2, 3 1, 4 – :

3	6	= −42 .
5	−4

) (k=3). 1

:

• 1, 3, 4 –

:		1	−1	0		= −4 ;
		3	−5	6
		5	−3	−4

•1, 2, 3 – -

:

	1	2	−1		= −28 ./
	3	4	−5
	5	−2	−3

0$ A k ,

k+1, $ , .

* A

A, . 1 .

81

>, . * -

rang A r(A).

& 1. +	1		2	−1	0
	A =	3	4	−5	6	.
		5	−2	−3	−4

,) A 3, $ ,

, . /

& 2. + B = (0	0	1 ).
0	0	0

, 1 1,

( b13), . /

2	0	2	2
& 3. - C = 4	4	8	16 .
8	4	0	2

, -

, , ,

				2			0	2
				4			4		8 . 1 :
				8			4		0
	A		= 2			4	8		+ 2		4	4		= −64 − 32 = −96 ≠ 0 .
						4	0				8	4

. 3.

+ Maxima,

5 rank.

/

1 n n, $

n . & , :

rang A= rang AT.

1 ( ) ' " * .

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

, $ .

82

& - A1, A2, …, An. ;,
- .
- . ) +. A1, A2, …, An 5
λ1, λ2, …, λn :
n
λ1 A1 + λ2 A2 + ... + λn An = λi Ai
i=1	$
6 ( ) - ' .,

5, , -

( ) 5 -

n

, : λ1 A1 + λ2 A2 + ... + λn An = λi Ai = 0 .

i=1

6 ( ) - ' ., $ -

n

λ1 A1 + λ2 A2 + ... + λn An = λi Ai ( ) -

i=1

, 5 λ1, λ2,…,λn .

6 ( ) - ' . ,

( ) ( -

) .

9 ( )		1	4	3
	A = 2		5	6	.
		3	5	9

, & A1,									A2,	A3. . λ = 3; λ = 0; λ = 1
												1	2	3
1		+ 0	4		3				+ λ A = λ A . .
: 3 2			5		= 1 6	, λ A
3			5		9		1	1	2	2	3	3

,

- ' .. /

1 ( ), $

. 0$ -, ( ) -

( ).

# : 9 ,

( ) ( -

).

+ -

. . , , -

36 . .

:

83

1);

2);

3),

, .

: & -

.

;

, $ -

, . - , ,

, .

r. . - r, -

, ) ! .

+ A =		1	3	6
	2		1	2	.
		3	4	8

,9 A , $

. ) ,

, 12 13 = −5 ≠ 0 . - ,

, .

* , , ,

, :						1	6	= −10 ≠ 0 .
						3	8
* ,
:		2	2		= −10 ≠ 0 . ) , -
		3	8

13 62 = 0 . /

# ) !- : 2 ( ) -

( ), $ .

2.16. 2 " !$" "’ $ 0 * * " *$. # 1 -1

64;1 , $ ’ , -

– , ’ .

& , ’ , ' -

n. ,

:

84

x1	+ x2 = 1,
2x1	+ 2x2	= 2,
3x1	+ 3x2	= 3

’ x1=2, x2=–1 ’ ,

:

x1	+ x2	+ x3 = 0,
{2x	+ 2x	2	+ 2x = 1
1			3

’ , .

* ( . . -

A *, $ -

:

	a11		a12	a1n			b1
A* =	a21		a22	a2n			b2		.
	a		a		a		b
		m1	m2		mn		m

+, $ A* -

A, .

., -

2-2.

# 1 -1 I ( ). 6 -

, A -

A*. 6

’ .

x1	+ x2 = 1,
" 2x1	+ 2x2		= 2, rang A = rang A* = 1 .
3x1	+ 3x2		= 3
x1	+ x2	+ x3 = 0,		rang A = 1; rang A* = 2. .
" {2x	+ 2x		+ 2x = 1
1	2		3

, 2- 2, .

# 1 -1 II ( ).

6 , $

: rang A = rang A* = n.

6 ’ . 0$ rang A < n,

’ .

= 2-2 ,

, , -

. & , $

’ , , ? ( . . 2.20). . ,

, $ ’ . 0$

, , $ ’ . 0$ ’ , , $ .

85

0

6
rang A = rang A*
		rang A < rang		A*
( ’ )
		( ’ )

9

rang A = rang A* = n

( ’ )

rang A = rang A* < n

( ’ )

1. 2.11. "

2-2

& 1. + 2-2 ,

’ :

2x		− 3x		+ 4x		= 3,
	1	2		3
x1	+ 5x2		− 4x3		= 2,

4x1 + 7x2 − 4x3 = 7.

,& l1 , l2 , l3 . + -

rang rang A*. ! , -

, :

2	− 3 4 3			2	− 3	4	3			2	− 3	4	3
1	5	− 4	2 l2 − l1 / 2	0	6,5	− 6	0,5		− 2l2	0	6,5	− 6	0,5 .
4	7	− 4 7 l3 − 2l1		0	13	−12	1	l3		0	0 0 0
. rang			= rang A* = 2, n = 3. . -

, ’ ./

& 2. + 2-2 ,

’ :
2x − 3x + 4x = 3,
	1	2	3
x1	+ 5x2	− 4x3	= 2,

4x1 + 7x2 − 5x3 = 7.

, + rang rang A*:

86

2	− 3 4 3			2	− 3	4	3			2	− 3	4	3
1	5	− 4	2 l2 − l1 / 2	0	6,5	− 6	0,5		− 2l2	0	6,5	− 6	0,5 .
4	7	− 5 7 l3 − 2l1		0	13	−13	1	l3		0	0	−1 0
. rang			= rang A* = 3 = n. . , -

’ ./

& 3. + 2-2 ,

’ :

2x1	− 3x2	+ 4x3	= 3,
x1 + 5x2 − 4x3 = 2,
4x1	+ 7x2	− 4x3	= 8.

, + rang
rang A*:
2 − 3 4 3			2	− 3	4	3	2	− 3	4	3
1 5	− 4	2 l2 − l1 / 2	0	6,5	− 6	0,5	0	6,5	− 6	0,5 .
4 7	− 4 8 l3 − 2l1		0	13	−12	2 l3 − 2l2	0	0 0 1
. rang		= 2, rang A* = 3. . .
9 ’ ./

2.17. "$ " , 2.5

6 ’ ,

.

9 I II 2-2 ? ( . .2.20 . 93)

' . 1’ 64;1.

3x − x

2

− 2 x

3

= 1,

2x + 4x + 3x = 7,

x

+5x

+ x

=1,

1

1

2

3

1

2

3

=5,

1.

2 x1

+ 4 x2

+

3x3 = 5,

−3x1 + x2

+ 2x3 =1,

2x1

+ 4x2 +3x3

5x1

+ 3x2

+ x3 = 2.

x1 + 5x2 + x3 = −1.

3x1

+9x2

+ 4x3

= 6.

x − 2x

+ x

= 1,

3x

+ x + 2x

=1,

2x

+ 3x

+ 2x = 2,

1

2

+

3

1

2

3

= 2,

1

2

3

2.

3x1

+ x2

2x3 = 3,

−x1

+ 2x2

− x3

3x1

+ x2 + 2x3 =1,

2x1

+ 3x2 + 2x3 = 1.

2x1

+ 3x2

+ x3

= 3.

5x1

+ 4x2 + 4x3 = 5.

−2x + x + 3x = 4,

−x + 3x − 2x = 2,

3x + x + 5x =1,

1

2

2x3

3

1

2

3

1

2

3

= 3,

3.

x1 − 3x2

+

= 4,

3x1

+ x2 + 5x3 =1,

x1

−3x2

+ 2x3

3x1 + x2

+ 5x3

= 2.

2x1

+ 4x2 + 3x3 = 3.

4x1 − 2x2 + 7x3 = 2.

3x − 4 x

2

+ 2 x

3

= 1,

3x − 4x + 2x = 1,

2x

+ 3x

+ x

=

1,

4.

1

1

2

+ x3

3

1

2

3

= 1,

2 x1

+ 3x2

+ x3 = 3,

2x1

+ 3x2

= 2,

x1

− 5x2

+ 3x3

4 x1

− 5x2

+ 3x3 = 1.

5x1

− x2 + 3x3

= 3.

8x1 + x2

+ 5x3

= 1.

−x + 3x +11x = 2,

x + x + x = 1,

x − 3x −11x = 1,

1

2

3

=1 ,

1

2

3

1

2

3

5.

−5x1 + 5x2

+ 2x3

−x1

+ 3x2

+ 11x3 = 3,

x1

+ x2 + x3 =

3,

x1 + x2 + x3 = 2.

x1 + 5x2 + 13x3 = 5.

3x1 − x2 − 9x3 = 4.

87

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

−x1 − x2 + x3 = 2.

x1 − 2x2 − 2x3 = 3,

7. x1 + x2 − 2x3 =1,x1 − x2 − x3 =1.

1 2 3

8.3x1 + x2 − 5x3 = −4,5x1 − 2x2 − x3 = −3.2x − 3x + 4x =1,

1 2 3

9.2x1 + x2 + 3x3 = 8,2x1 + 3x2 + 2x3 = −2.x − 2x + x =10,

1 2 3

10.2x1 − x2 − 2x3 = −1,5x1 + x2 − x3 =1.3x + 2x + x = 2,

1 2 3

11.3x1 + 4x2 − x3 = 3,

4x1 + 5x2 − 2x3 = 8.2x + 3x + 4x = 5,

1 2 3

12.2x1 − x2 − 3x3 = −7,5x1 − 4x3 =1.3x + x − x = 3,

1 2 3

13.5x1 + x2 + 3x3 = 2,8x1 + 2x2 + x3 = 6.3x + x − 2x = 4,

	2x − x + 3x = − 4,
		1		2	3
14.	x1	+ 3x2			− x3	= 2,
	5x1 + 2x2 + x3 = 5.
	3x − 2x + 4x = 5,
15.		1	− x2		2		3
	2x1				− x3	= 4,
	x1 + 5x2				− x3	= 2.
	3x + 4x − 4x = 5,
		1	+ x2		2		3
16.	2x1				+ 4x3		= 3,
	5x1		+ 5x2 − x3				= 0.
	7x + 5x + 2x = 9,
17.		1			2		3
	x1	− x2		− x3 = 3,
	x1	+ x2		+ 2x3 = 2.

1 2 3

18.−4x1 + 7x2 − x3 =10,−4x1 + x2 + 5x3 = 3.8x − 5x − x = 5,

−2x + x			+ x = 2,
	1	2		3
x1	+ x2	− x3 =1,
−x1 + 2x2			= 3.
x	− x	− x = 2,
1	2		3	=1,
x1	+ x2	− 2x3
31	+ x2	− 5x3		= 4.

3x		+ x		− 5x = 5,
	1	2					3	= −1,
2x1		− 3x2				+ 4x3
5x1		− x2 + 3x3 =1.
2x + x + 3x = 6,
	1	2					3	= −3,
2x1		+ 3x2				+ 2x3
4x1		+ 4x2				+ 5x3		= 3.
2x − x − 2x =1,
	1	2					3
3x1		+ 2x2			+ x3 =1,
2x1		+ 3x2			+ 3x3 = 2.
3x + 4x							− x	= 2,
	1			2			3		= 3,
2x1		+ 3x2					+ 4x3
5x1		+ 7x2					+ 3x3		= 5.
2x − x − 3x = 2,
	1		2				3
3x1		+ x2			− x3 = −1,
5x1		− 4x3 =1.
3x + x − 2x = 4,
	1		2				3	= 9,
2x1		− 3x2					+ x3
5x1		+ x2			+ 3x3			= −4.
2x − x + 3x = 4,
	1		2				3
x1	+ 3x2				− x3 = 3,
4x1 + 5x2 + x3 = 10.
x + 5x					− x = 2,
1			2				3		= 5,
3x1		− 2x2					+ 4x3
4x1		+ x2			+ 3x3 = 7.
2x + x + 4x = 7,
	1		2				3		= −8,
3x1		+ 4x2					− 5x3
5x1		+ 5x2					− x3	= −1.
x − x			− x = 3,
1		2	+				3
x1	+ x2					2x3 = 2,
2x1 + x3 = 5.
8x − 5x − x = 5,
	1			2			3		= 3,
−4x1 + x2							+ 5x3
4x1 − 4x2							+ 43 = 8.

2x − x − x =1,

1 2 3

−x1 − x2 + x3 = 3,

x1 − 2x2 = 5.

x	− 2x	− 2x = 2,
1	2	3
x1	− x2 − x3 = 0,
3x1 − 4x2 − 4x3 = 5.

2x		− 3x			+ 4x				= 0,
	1		2				3
5x1		− x2 +				3x3 = 2,
7x1		− 4x2 + 7x3 =1.
2x + 3x + 2x = −1,
	1			2			3
x1	− 2x2			+ x3 =					8,
3x1 + x2				+ 3x3 = 5.
2x + 3x + 3x = 2,
	1			2			3
3x1		+ 2x2 + x3						= −1,
−x1		+ x2 + 2x3						=1.
4x		+ 5x			2	− 2x = 3,
	1								3	= 4,
2x1		+ 3x2				+ 4x3
6x1		+ 8x2				+ 2x3				= 1.
2x − x − 3x = −9,
	1		2		+ x3		3
x1	+ 2x2						= 3,
3x1 + x2					− x3		= 2.
2x − 3x						+ x			= 9,
	1			2			3		= 4
3x1		+ x2			− 2x3,
5x1		− 2x2				− x3			= 3.
5x + 2x						+ x			= 5,
	1	− x2			2		3		= 2,
2x1					+ 3x3
7x1		+ x2			+ 4x3				= 3.
2x − x					− x		= 4,
	1		2			3	= 2,
x1	+ 5x2				− x3
3x1 + 4x2 − 2x3 = 0.
2x + x					+ 4x				= 7,
	1		2				3		= 3,
2x1		− x2			− 3x3
3x1		+ 4x2 − 5x3 = −8.
x + x			+ 2x				= 2,
1		+	2			3				= 9,
7x1			5x2			+ 2x3
8x1		+ 6x2				+ 4x3				= 5.
−4x + x + 5x = 3,
		1			2				3
8x1		− 5x2				− x3 = 4,
4x1		− 4x2				+ 4x3 = 3.

88

1 2 3

19.4x1 + 2x2 = 3,

x1 + x2 + 2x3 = −3.3x + 2x − 5x = 2,

1 2 3

20.3x1 + 4x2 − 5x3 = 2,5x1 + 5x2 − x3 = 6.2x + x + 4x = 5,

1 2 3

21.2x1 − x2 − x3 = 5,3x1 − 2x2 + 4x3 = 7.x + 5x − x = 7,

1 2 3

22.x1 − 2x2 − 2x3 = 3,2x1 − 5x2 − 4x3 = 4.x + x − 2x = 1,

	7x + 5x				+ 2x = 14,
23.		1	2			3
	x1	− x2	− x3			= −1,
	x1	+ x2	+ 2x3 = 4.
	2x + x		2	− x = 2,
24.		1				3
	x1	+ x2	+ x3			= 6,
	3x1 − x2 + x3 = 4.

1 2 3

25.2 x1 − x2 − 3x3 = −1,

x1 + 5x2 + x3 = 6.3x + 4 x + 2 x = 8,

2x1 − x2 + 2x3 = − 4,

26. x1 + x2 + 2x3 = −1,x1 − 2x2 = −3.

1 2 3

27.2x1 − x2 − 6x3 = −1,3x1 − 2x2 = 8.x + x + x = 2,

	x		+ 5x		2		+ x		= 0,
		1						3
28.	3x1 + 4 x2 + 2 x3 = 8,
		−2 x + x					2	− x	3	= 5.
				1
	2x1 − x2 − x3 = 0,
29.			+ 5x2				+ x3 = −7,
	x1
	x		− 2x				− x		= 2.
		1			2			3
	3x1 − x2 + 5x3 = 7,
30.			+ 2 x2				+ 3x3 = 6,
	x1
		2 x − 3x				2		+ 2x = 1.
			1							3
	2 x1 − x2 − x3 = 4,
31.
	3x1 + 4 x2 − 2 x3 = 11,
	3x			− 2 x		2		+ 4 x		3	= 11.
		1

4x + 2x = 3,
	1				2	− 5x3 = 2,
3x1		+ 2x2
x1 + 5x3 =1.
2x		− x		2	− 3x				= 3,
	1						3			= 7,
2x1		+ x2			+ 4x3
4x1		+ x3			= 10.
2x		− x			− x			= 3,
	1			2			3	= 5,
x1	+ 5x2				− x3
3x1 + 4x2 − 2x3 = 8.
x − 2x − 2x = 3,
1	+ x2			2				3
x1				− 2x3				= 0,
x1	− x2			− x3 =1.
7x		+ 5x				+	2x				= 14,
	1				2					3
x1	− x2 − x3						= −1,
8x1 + 4x2						+ x3 = 13.
x + x			2	+ x			= 6,
1						3		= 4,
3x1		− x2			+ x3
4x1		+ 2x2 =					10.
3x		+ 4 x			2	+	2 x			3	= 8,
	1
x1	+ 5x2				+ x3			=		6,
2 x1 − x2					+ x3			= 2.
x + x				+ 2x				= −1,
1			2				3			= −4,
2x1		− x2			+ 2x3
4x1		+ x2			+ 4x3					= −2.
3x		− 2x			2	=	8,
	1						= 2,
x1	+ x2 + x3
2x1 − 3x2 − x3 = 6.

3x1 + 4 x2 + 2 x3 = 8,
			− 3x3 = −1,
2 x1 − x2
x	+ 5x	2	+ x = 0.
1			3

x	+ 5x	+ x = −7,
1	2		3
2x1 − x2 − 3x3 = 0,
	1	2	= 2.
3x + 4x
5x1 + 8x2 − x3 = 12,
	+ 2 x2	+ 3x3 = 6,
x1
2 x − 3x		2	+ 2x = 1.
	1		3

3x1 − 2 x2 + 4 x3 = 11,
					= 4,
2 x1 − x2 − x3
x	− x	2	+ 5 x	3	= 7.
1

x

+ x

−

2x

= −3,

1

2

3

= 2,

3x1

+ 2x2

− 5x3

4x1

+ 3x2

− 3x3

= 7.

2x

+ x

2

+ 4x

= 7,

1

3

= 3,

2x1

− x2

− 3x3

3x1

+ 4x2 − 5x3 = 2.

3x

− 2x

+ 4x

= 5,

1

2

3

x1

+ 5x2

− x3

=

7,

2x1 − 7x2 +

5x3 = 1.

x

− x

− x

= 2,

1

2

−

3

= 0,

x1

+ x2

2x3

2x1 − 3x3 = 5.

2x − 2x − 2x = 3,

1

2

3

4,

x1

+ x2 +

2x3

=

3x1 − x2

= 5.

3x

− x

2

− x

= 4,

1

3

= 2,

2x1

+ x2

− x3

x1 − 2x2

= 1.

2 x

− x

2

− 3x

3

= −1,

1

= 8,

3x1

+ 4 x2

+ 2 x3

5x1

+ 3x2

− x3 = 2.

2x − x + 2x = −4,

1

2

3

= −2,

4x1

+ x2

+ 4x3

−2x1 −

2x2 − 2x3 = −8.

2x

− x

− 6x

= −1,

1

2

3

3x1

− 2x2

= 8,

− x1

+ x2

− 6x3 = 5.

2 x1 − x2 − 3x3 = −1,

+ 4 x2 + 2 x3 = 8,

3x1

5x

+ 3x

2

− x

= 7.

1

3

x

− 2 x

2

− x

= 2,

1

3

x1 + 5x2 + x3 = −7,

1

+ 3x

2

= −5.

2 x

x

+ 2 x

2

+ 3x

= 6,

1

3

2 x1 − 3x2 + 2 x3 = 1,

3x

− x

2

+ 5x

= 2.

1

3

2 x1 − x2 − x3 = 4,

+ 4 x2 − 2 x3 = 11,

3x1

5x

+ 3x

2

−

3x

3

= 7.

1

89

2.18. ! ! ". 3$ ! !. / *- "’ ! !

6 m n -

, $ . .

:

a	x + a		x + a	x +... +	a x =0,
11	1 12		2 13 3		1n	n
a21x1 + a22 x2 + a23 x3 +... + a2n xn						=0,
						.
,
a	x + a		x + a	x +...+ a	x	=0
m1 1		m2 2		m3 3	mn n

6 , $

				0
, , ( ) ’	X	0	=	0		. " -
				0

: A X = 0 ,

– 5 , .

0$ m = n, ,

’ , 5 2. !, 5 2 ’

5 x j =			Aj			(j=1, 2, …, n). + –
				A

A ≠ 0 , – , -

j (

). - -

, – Aj -, , , ’ .

x + 3y − z = 0,

+ ’ 64;1 2x + 2 y + z = 0,

4x + 11y − z = 0

,∆ =		1	3	−1		= −9 ≠ 0 , x = y = z = 0./
		2	2	1
		4	11	−1

# . ! , $ -

’ , , $ < : 0.

- , $ < : 0, ’ . 0$ <=0, ’ .

90