Высшая математика. Том 1

3.	A1 (7, 2, 4), A2 (7, −1, −2), A3 (3, 3, 1), A4 (−4, 2, 1).
4.	A1 (2, 1, 4), A2 (−1, 5, −2), A3 (−7, −3, 2), A4 (−6, −3, 6).
5.	A1 (−1, −5,	2), A2 (−6,	0,	−3), A3 (3,	6, −3), A4 (−10,	6, 7).
6.	A1 (0, −1, −1), A2 (−2, 3, 5), A3 (1, −5, −9), A4 (−1, −6, 3).
7.	A1 (5, 2, 0), A2 (2, 5, 0), A3 (1, 2, 4), A4 (−1, 1, 1).
8.	A1 (2, −1,	−2), A2 (1,	2,	1), A3 (5, 0,	−6), A4 (−10, 9,	−7).
9.	A1 (−2, 0, −4), A2 (−1, 7, 1), A3 (4, −8, −4), A4 (1, −4, 6).

10.	A1 (14,	4,	5),	A2 (−5, −3,				2),		A3 (−2,		−6,		−3), A4 (−2,			2,		−1).
11.	A1 (1,	2,	0),	A2 (3,		0, −3),				A3 (5,		2,	6),		A4 (8,	4,		−9).
12.	A1 (2,	−1,	2),	A2 (1,		2,		−1),		A3 (3,		2,		1), A4 (−4,			2,		5).
13.	A1 (1,	1,	2), A2 (−1,			1,		3), A3 (2,				−2,		4), A4 (−1,			0,		−2).
14.	A1 (2,	3,	1),	A2 (4,		1, −2),				A3 (6,		3,	7),		A4 (7, 5,			−3).
15.	A1 (1,	1,	−1),	A2 (2, 3,				1), A3 (3,				2,	1),		A4 (5,	9,	−8).
16.	A1 (1,	5,	−7),	A2 (−3, 6,				3),		A3 (−2, 7,				3), A4 (−4,			8, −12).
17.	A1 (−3,	4,	−7), A2 (1,			5,		−4),		A3 (−5, −2,					0), A4 (2,		5,		4).
18.	A1 (−1,	2,	−3), A2 (4,				−1,		0),		A3 (2,		1,	−2), A4 (3,				4,	5).
19.	A1 (4,	−1,	3),	A2 (−2, 1,					0),	A3 (0, −5,				1), A4 (3,			2, −6).
20.	A1 (1,	−1,	1),	A2 (−2, 0,					3),	A3 (2,		1,		−1), A4 (2,			−2, −4).
21.	A1 (1,	2,	0),	A2 (1, −1,				2),		A3 (0,		1,	−1), A4 (−3,				0,	1).
22.	A1 (1,	0,	2),	A2 (1,		2, −1),				A3 (2,		−2,		1), A4 (2,		1,		0).
23.	A1 (1,	2,	−3),	A2 (1, 0,				1),		A3 (−2,		−1,		6), A4 (0,			−5, −4).
24.	A1 (3,	10,	−1),	A2 (−2, 3,					−5),		A3 (−6,		0,		−3), A4 (1,			−1, 2).
25.	A1 (−1,	2,	4),	A2 (−1, −2,					−4),		A3 (3,		0,		−1), A4 (7,			−3,		1).
26.	A1 (0,	−3,	1),	A2 (−4, 1,					2),	A3 (2,		−1,		5), A4 (3,			1,		−4).
27.	A1 (1,	3,	0),	A2 (4, −1,				2),		A3 (3,		0,	1),		A4 (−4, 3,			5).
28.	A1 (−2,	−1, −1),			A2 (0,		3,		2),		A3 (3,		1,	−4), A4 (−4,				7,		3).
29.	A1 (−3,	−5, 6),			A2 (2,	1,		−4),		A3 (0,		−3,		−1), A4 (−5,				2, −8).
30.	A1 (2,	−4,	−3),	A2 (5, −6,					0),	A3 (−1, 3,				−3), A4 (−10,				−8, 7).
31.	A1 (1,	−1,	2),	A2 (2,		1,		2),		A3 (1,		1,	4),		A4 (6,	−3,		8).

141

3.22. & 1 &

a11	a12
" A = a21	a22
a	a
31	32

a13	!	x1	!
a23	##	- X = x2	## ,
a	%#	x	%#
33		3

) ) A.

% X:

AX = λ X ,

! – . &, - !

’ :

	0	!
X =	0	## = 0 .
	0 %#

* !, ’ ,

A, - X !

A.

& ,

# , ,

. % -

. % ' , -

) )) .

3.23.0 # & ! & 1 &

& A. ' E· X = X, -
AX = λ EX			( A − λ E ) X = 0 .
a11 − λ		a12	a13	!
A − λ E =	a21	a22 − λ	a23	## .
	a	a	a − λ %#
	31	32	33

%-

.

	(a11 − λ ) x1 + a12 x2 + a13 x3 = 0,
			+ (a22 − λ ) x2			+ a23 x3 = 0,
a21x1
a x			+ a x	2	+ (a	− λ ) x = 0.
	31	1	32		33	3

',

x1, x2, x3 X. 1 ’

,					, :
	a11 − λ	a12	a13
	a21	a22 − λ	a23	= 0 ,		A − λ E		= 0 .
	a31	a32	a33 − λ

142

. !. -

A !. ( ! X,

!.

! , -

. % -

:

	λ1	0		0
	0	λ2		0
							≡ A .
	0	0	λn

# 1. & ) -

A = 1 1!# . & # , )) -

8 3%

.

4 +		1 − λ	1		= 0 .
		8	3 − λ

, , -

:

−(1 − λ )(3 − λ ) − 8 = 0. & −λ 2						− 4λ − 5 = 0.			λ = −1; λ = 5 .
0			2	0			2		1	2
# ’ !1		= –1.
# !1 = –1
:	2x + x = 0			2x + x = 0				2x + x = 0,
			&				&
	1	2			1	2			1	2
8x1 + 4x2 = 0				2x1 + x2 = 0					0 = 0.

' ,

– . )

) ( 1),

( 2), :

rang A = rang A* < n.

& ’ :

	x1 = C				C	!
x	= −2x = −2C	&	X1 =		−2C	# , R.
2	1					%
/, =1 ’ X1							=	1 #! .
								−2%

143

$ ’	!2 = 5.
−4x1 + x2 = 0			x1 = C	C !	R.
8x − 2x = 0 &		x = 4x = 4C & X2 =		4C # ,
1	2	2	1	%
/, =1 ’ X2 = 1 #! .
				4 %
& Maxima. $ -
				matrix -

, .

1 # 1 , eigenvalues. . , , . # – A, – -

.

9 eigenvalues , solve,

’ . 6 , solve

’ , .

1 ,-eigenvectors. . , ,

– , -

– , .

. 3.31. #

/ , Maxima

0: λ1 = 5; λ2

= −1, : X1

1 !

1 !

=

# ,

X2 =

# .

4 %

−2%

/ # , )) -

=

−1

0

.5

: A

0

5

# 2. & ) -

1

−1

0

!

2

1

#

A =

−1# .

−1

#

1

0 %

144

4 + :

	1 − λ	−1		0
	2	1 − λ		−1		= 0 .
	1	−1		0 − λ
:
(1 − λ ) 1 − λ −1	+ 1 2 −1		= (1 − λ )(λ 2 − λ − 1) − 2λ + 1 =
−1 −λ	1	−λ

= λ 2 − λ − 1 − λ 3 + λ 2 + λ − 2λ + 1 = λ (−λ 2 + 2λ − 2) = 0 .

' ’ (

D = b2 − 4ac = 22 − 4 2 = −4 < 0) ’ λ = 0 .

# ,

:

	x1 − x2 = 0					x1 = x2 = C		C !
		+ x	− x = 0				= 3x = 3C &	X =	C # .
2x				& x
	1	2	3		3		2		#
	x1 − x2 = 0						C R		#
									3C %

1!

/, =1 ’ X = 1## . 5

3#%

3.24. " $ (" ! %)

! ) ,

, -

" $ ( ) ).

" n ) S1, S2,…, Sn )

x1, x2, …, xn. # , ij -

, ) Sj ) Si. 2-

, -

), ), :

n
3aij = 1(j=1, 2, …, n).
i=1
a11	a12	a1n !
# = a21	a22	a2n ## ,
		#
an1	an2	ann %

% . ,

n

3aij = 1, - # 1.

i=1

$ -) ) Si (i=1, 2,…, n) ) )

145

:

pi = ai1x1 + ai2 x2 + + ainxn .

$ ) , )

) Si, ) ) ))

:

pi ≥ xi (i=1, 2,…, n).

! , pi ≥ xi (i=1, 2,…, n), -

:

a11x1 + a12 x2 + + a1n xn ≥ x1,

a21x1 + a22 x2 + + a2n xn ≥ x2 ,

,

an1x1 + an2 x2 + + ann xn ≥ xn.

+ ) , :
	x1 (a11 + a21 + + an1 ) + x2 (a12 + a22 + + an2 ) +
	+ xn (a1n + a2n + + ann ) ≥ x1 + x2 + + xn .
	n
& 3aij		= 1, ,
	i=1	x1 + x2 + + xn ≥ x1 + x2 + + xn .
) :
/ ,			pi ≥ xi (i=1, 2,…, n) ,					pi ≥ xi
pi	= xi (i=1, 2,…, n). & ) ,
) .
		x	!
		1	#
% X = x2			# ), -
		#
		xn %
: AX = X , AX = λ X ,
#,							λ = 1.
+ ) S1, S2, S3 :
			1/ 3	1/ 4	1/ 2	!
			# = 1/ 3 1/ 2 1/ 2			# .
			1/ 3	1/ 4	0	%#
& ) ) .
4 & ,								λ = 1.
$ ’ (# – λ /) =0.
1/ 3 − 1	1/ 4	1/ 2 !	−2 / 3 1/ 4				1/ 2 !
1/ 3	1/ 2 − 1	1/ 2 # = 0,		1/ 3	−1/ 2		1/ 2 # = 0 .
	1/ 4	#			1/ 4		#
1/ 3		0 − 1%	1/ 3				−1 %

146

’ 8 : x1 = (3/ 2)C ,

x2 = 2C , x3 = C .

' , )

x = (3/ 2C;2C;C ),

) 3 : 4 : 2. 5

3.25. , " # - 3.7

+ ’ , ))

) .

& #. &

# , )) .

1.	−	2	−2.
	4			5
	010		−72
4.	−	6			5 .
	4				5
	0−14,4			−112
7.	−	1		5 .
	4				5
	0−2,4			−62
10.	−	3	5 .
	4			5
	0−6		−82
13.	−	2		5 .
	4				5
	0−3,6			−72
16.	−	5			4 .
	4				5
	0−13,5			−102
19.	− −3			4 .
	4			5
	00,5		−22
22.	−2		−4.
	4			5
	05		−72
25.	−−10 −28.
	4	2		5	5
	0				2
28.	−−7		5.
	4	−4		5
	0		22
31.	−	5		2 .
	4				5
	0−28			−102

2.	−−7
	4	−4
	0
5.	−	5
	4
	0−28
8.	−	2
	4
	0−4
11.	−	3
	4
	0−5,6
14.	− −7
	4
	0−3,6
17.	−	3
	4
	0−7
20.	− −3
	4
	0−0,5
23.	−−3
	4	3
	0
26.	−−13
	4	2
	0
29.	−−2
	4	0
	0

5.

5

22

2 . − 5

102

5.

−5

72

5 . − 5

82

5.

5

22

4.

−5

82

−4. − 5

22

0.

−5

22

−40. 5

5 2

3.

−5

32

3.	−	−2	3 .
	4	0		5
	0		−32
6.	−	5		2 .
	4			5
	0−40		−132
9.	−	−2	5 .
	4	0		5
	0		−32
12.	−	1	5 .
	4			5
	0−2		−62
15.	−	6		4 .
	4				5
	0−17,5			−112
18.	−	−5	4.
	4	−1	5
	0		02
21.	−	−7	10.
	4	−2		5
	0		2 2
24.	−−11 −14,4.
	4	5		6	5
	0				2
27.	−	2	−2.
	4			5
	010		−72
30.	−	6		5 .
	4				5
	0−14,4			−112

147

3.26. 1 1 )

# ’ -

,.

# 1. ( , L( 1, x2, … , xn) n -

, ) ) ,

, ,:

n n

L(x1, x2 , , xn ) = 33aij xi xj .

i=1 j=1

#, , ) , – , -

aij = aji . ", . 7

# = ( ij) (i, j = 1,2,…, n), ,, %

0. :) ,

, – , -

) ,.

% , :

x

L = X T AX ,

!

1

#

x2

#

- ,

T

X =

# –

X = ( 1, 2, ..., n) – -

#

xn %

.

- XT

1 x n -

$ ,

) , #

n x n

- -

1 x n,

- X n x 1

,

1 x 1:

a11 a12

a1n ! x1

!

L = ( x x

x

a

a

a

# x

#

)

21

22

2n #

2

# =

1

2

n

#

#

1xn

#

#

an1 an2 ann % xn %

nxn

nx1

x1

!

n n

x

n n

x

n n

x

!

x

#

=

=

3 3 a

j1

3 3 a

3 3 a

#

2

#

i

j2 i

jn i

#

#

i=1 j=1

i=1 j=1

i=1 j=1

%

#

xn %

n n

n

n

n n

n

n

= 3 3 a j1xi x1 +

3 3 a j2xi x2 + +

3 3 a jnxi xn = 3 3 aij xi x j .

i=1 j=1

i=1 j=1

i=1 j=1

i=1 j=1

148

$	, L(x , x , x			) = 4x2	− 12x x		− 10x x		+ x2	− 3x2 .
	1	2	3	1	1	2	1	3	2	3
& )) .
4 &	) ,. ' -
)	, , 4, 1,

–3, – , ) ,,:

L = ( x			4 −6 −5! x1					!
	, x	, x	)	−6	1	0	# x	#	.5
1	2	3					# 2	#
				−5	0		#	#
						−3% x3 %

&' , , -
.
x1		!	y1			!
x		#	y			#	'
" - X =	2	#	Y =		2	#
		#				#
	#					#
xn %			yn %

X = 1, = ( ij) (i, j = 1,2,…, n) – -
n- . / , L = X T AX :
L = (CY )T A(CY ) = (Y T CT ) A(CY ) = Y T (CT AC )Y , L = Y T A*Y , A* = CT AC .
/	,
(CY )T = (Y TCT )( . «- -
»).	X = CY
',

) , : A* = CT AC .

$ , L( x , x )			= 2x2	+ 4x x				− 3x2 . & -
, L( y1, y2 )	1	2	1			1	2			2
			)
x1 = 2 y1 − 3y2 ; x2 = y1 + y2 .
4 7-			x1	!				y1		!	'
		X =		#		Y =				#
			x					y
			2 %						2 %
	2	−3!	2- .
X = CY , C =		#
	1	1 %
7 ) ) , A =				2			2	!
								# . 7 ) -
				2		−3%

149

) , L( y , y			) = Y T A*Y .
	1	2
&			) ,:
A* = CT AC =	2	−3!T 2			2 ! 2 −3!
	1	1 %# 2			−3%# 1 1 %# =
2 1! 2	2 ! 2			−3! 6 1 ! 2		−3! 13 −17 !
= −3 1%# 2	−3%#		1	1 %#	= −4 −9%# 1	1 %#	= −17 3 %# .

' ) ) ,

, , –

, ) ,, :

L( y1, y2 ) = 13y12 − 34 y1 y2 + 3y22 . 5

3.27. 1 % 1 )

+ ,

.

n n

( , L = 33aij xi xj ( -

i=1 j=1

), )) , aij = 0 i ≠ j , -

) , :

n

L = a11x12 + a22 x22 + + ann xn2 = 3aii xi2 . i=1

+ .

. 2- , -

.

3.28. + 1 ) 1 % %

& ,

, , , -

. 0 , , ,

) ,, ) ,

.

( ) , ,

,

. 2

: - .

0 % !
( ) ) ) ,	L(x1 ,x2 , ,xn ) .	#
( x1 ,x2 , ,xn ) ( y1 ,y2 , ,yn )		-

150