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Национальный минерально-сырьевой университет «Горный»

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% 4 A, A A 1.1 1.2 A (1.36) A,

A 2, H ' A 4 2 5, A

D 5 7, 7 A 1.2.

2 23 A D = 0 , A = A A 4 ' 3 2 ' A, 2 ' = (2, 2I, A

A A, D 2 ', D66 = 2I). , A A ' 2 7.

1) r = 2.

@ , A 7 A = A 3

7 ' , 2 27.

% 3, A, 2I A D23 D2 A a23

D	=	a11	a12	0.	(1.38)
23		a31	a32		(1.38)

0 D A 2 A (1.36) D 2 A

- 3:

a11x1 + a12 x2 + a13 x3 = 0,

+ a

x = 0.

(1.39)

31 1

A 5 7 A (1.39), A 7 A

23 x1 x2

2 2 A, A D23 :

a11x1 + a12x2 = a13 x3,

a x

+ a

= a

x .

(1.40)

31 1

2 23 D A

D23 0 ,

2 23, A # A

4-A 23 7 A

x3 ("'"

, 2' = ' 2), A

a13x3

a12

a11 a13x3

a33x3

a32

;

a31 a33x3

D23

" 23 7 2 2, A 2 4 7.

a13x3

a12

= x

a13

a12

= x

a12

a13

= x D ,

a x a

3 21

a11 a13x3

= x

a11

a13

= x D .

% 7 2 ' A 2 7A

A21 = ( 1)2+1 D21 = D21; A22 = ( 1)2+2 D22 = D22 ;

A23 = ( 1)2+3 D23 = D23 ,

A 2 A 5 7A A (1.40) ' 2 ' :

x =	A21	x , x	2	=	A22	x .	(1.41)

1	3			3
	A23				A23
+ A 2 (1.41) 27I ' A 4 5 A
(1.39), 4 «'» . x3 .							% 4 A
A 23	7 A			x3			A ' 3

2 6 A 2 A (1.41) IH 7 x1 x2 . % 2 A ' A 2 7 5

A (1.39) ' A 4 A 4, 27 A

6 A 2 A (1.41).

9. % 2 A A

' 5 7 A 2 A7 A. 2) r =1.

, A = A (1.36) =, A

D2 A D A = 2I, 7 ' D2 A A = -

D66 = A (1.36) - 2I.

% 3, A, a11 0 . 0 D A 2 A (1.36)	D 2
A I
a11x1 + a12x2 + a13x33 = 0.	(1.42)
% D A 2 ' 3 7, 2I' 5 7 (1.42) 7 27 7 4
5 A 3 A (1.36).
, A 7 (1.42) A A x1 ,

A, 23 7 7 x2 x3 ,
A 5:				a12				a13
x		=			x	2			x .	(1.43)

1				a11			3
								a11
+ A 2 (1.43) ' A 4 5 7 (1.42).
2: 23 7	x2		x3				2			6 A 2 (1.43)

IH A x1 - 27I 5

7 (1.42) ' A 4 A 4, 27 A

6 A 2 (1.43).

9. % 2 A A

' 5 7 2 7 A7 A.

2 2 A 2 , 2

2 . A 2 A, 27IH

7 3, A 2 7 A 2 . 0 2 A A 2 2 A 4 3 3 2 5 3 '

4 7 D 5. 0 A A 2 4

A # A 6 A 2 # A. /2 A A D 6 A 2 7 27 7 7 4 27 5 7 2 A D66 = D A. 23 2 # A 7

3A A A 2 7A. 0 D A 5 ' 2 ' A

7 27 7 A / - A 2 23 2I 7 ,

2 23 A A.

A 'H 2 A 23 2 2

23 A 2 A - A m 2 n

A, ' IH ' 2 2 A =, A

[1].

6 K 5

9 1. , 5 3 A

2x1 3x2 + x3 = 11,

x1 + 5x2 4x3 = 10,
	4x	+ x		3x	= 5.
	1	2		3
6 K. 0 2 A 2 23 A
	D =		2	3	1	= 2.
			2	3	1
			1	5	4
			4	1	3

2 23 A 2 27, 2 23, A

A A 5, 27 A 6 A 2 A # A. 0 27 A 2 2 D1, D2, D3 .

				D1 =		11	3			1		= 6;						D2 =		2		11		1	= 2;
						11	3			1										2		11		1
						10	5 4													1 10 4
						5	1		3											4		5 3
								D3 =					2			3		11	= 4.
													2			3		11
													1		5			10
													4		1			5
6 A 2 A # A																							D3
x	=	D1	=	6	= 3;		x		=		D2			=		2	= 1;				x	=	D3	=		4	= 2.
		D1

1		D		2			2				D					2					3		D			2
		D		2							D					2							D			2
9 2. " 2 3 A
							x			2x					+ x		= 1,
									1	2					3
							3x1 + x2 + 4x3 = 3,
							x			+ 5x					+ 2x			= 2.
									1	2					3

6 K. 0 27 A 2 23 A:

1 2 1

D = 3 1 4 = 0.

15 2

D = 0, A 2 ' 2, 2 ' A. A D1.

D1 =	1	2	1
	1	2	1
	3	1	4	0.
	2	5	2

23 2 2 A 4 4 27 3, , D1 2 27, 2, A A.

9 3. , A A A
x	+ x	2	2x			= 1,
1		2			3
2x1 + 2x2 4x3 = 2,
3x + 3x				2	6x = 5.
	1			2		3
3 D = 0, D1 = 0, D2 = 0, D3 = 0.						A A, A 2

' 3 7, A 4 ' 7 3. % 2 A

3x1 + 3x2 6x3 = 3 ,							3 A I A,
2 23, A A 5.
9 4. , A A A
x		x		+ 3x		= 1,
	1		2		3
2x1 2x2 + 6x3 = 2,
3x			+ 4x		x		= 4.
	1			2	3
3 D = 0, D1 = 0, D2 = 0, D3 = 0.						2 7

A 4 A ' 7 2, 7 A

23 A A ' A 4 5,

3 2.

9 5. 5 7 A

3x1 x2 + 2x3 = 0,2x1 + 3x2 4x3 = 0,5x1 2x2 + x3 = 0.

6 K. ! 7 A 7 27 7 A 2

. 0 2 A 2 23 D A

D =	3	1	2	= 31.
	3	1	2
	2	3	4
	5	2	1
D 0 , A A 2 5:
x1 = 0;		x2 = 0;		x3 = 0.
		43

9 6. " 2 3 A 2

5 7.	+ 3x		+ 2x	= 0,
x	+ 3x		+ 2x	= 0,
1		2	3
2x1 2x2 + x3 = 0,
x		5x	x	= 0.
1		2	3
6 K. 0 2 A 2 23 A
		1	3	2
	D = 2		2	1 = 0.

15 1

D = 0 , A, A 2 5 7, A ' A 4 2 5. A 5 7 A. 27 A

A = A, 27 2 A A, A D11 :

D		=	2		1	= 7.
D		=	2		1	= 7.
11				5	1
				5	1
D11 0 , A = A r = 2							, 2 23, A
D 2 A - 3. , 5 A
A, 7H I					3		, 23
x2 x3 :
2x2 + x3 = 2x1,
	5x			x	= x .
		2		3	1

2 23 D A D11 0 , 2 23,

A # A

4 A 23 A x1 A A 5:

2x1

3x1

, x =

= 8x1 .

D11

A ' A, A 4 5 A A 4 3

x2 = 73 x1,

x3 = 87 x1,

x1 - 2I' H 2.

4 5 A ' ' 3

A 6 A, 4 7 A

23 A t . 2 5 A 2 7 3 x71 = t , 5

A A 4 3

x1x2x3

= 7t,

= 3t, t .

= 8t,

8 % ! # 1.2

1.0 A 2 A n 2 n A A

2.% A 2 A n 2 n A

A 4 ' 3 A 2 2?

3.6 A 2 A # A 5 6 A 2 # A 27

A n 2 n A.

4.0 A 2 2 7 A?

5.4 2 A 2 ' 3 A?

6.6 A 2 ' A 2 7 2 7 2

5 A n 2 n

7.7 A A = n - 7?

8.# A H3I 7 7 A = A 5 7

2 A, D 2 A m 2

n A?

1.3. % 4 K : ! %4 & #

% A 0 A A 3 7 2 IH A A:

•8 % .

•6 # # ! % .

•$* .

•,' .

•6 K % ! %4 & # L .

% 2 7 A 2 0 A 2 3 27 A 5 3 .

2 0 ' 3 7 , ' 3 [1], 2 2, . 45-75 2 2 I – A 2 I A 2 4.

A - 6 A ' 7 5 3

23 ' Y1 A A Y 6-10.

8 %

, A 23 2 7 A =, ' H 7 3 4 5

2 A = - 2 2I A =. 3

A A 3 7 A 7A A = A, A IH A 5 5 2 4 7, ' 2 7 A 7

A 3I .

2 A = 7 XIX ,

7 4 A A A 2 7, 5 '2 A

A . 3 A A A 5 A

2 7.
, !.	% 7A 23 7	'2 =	2

aik (i = 1, 2,..., m;k = 1, 2,..., n), 7H 7 m n 2'=, 7

5 m ×

n ' 7 :

a11

a12

...

a1n

a21

a22

...

a2n

(1.44)

...

am1 am2

...

amn

aik

(i = 1, 2,..., m;

k = 1, 2,..., n).

A, B

A = ' I 2 ', A A =

. . 2, ' IH A =, I 7 M!. # 4 D2 A

aik

A ;

i ' A ,

k - A 2'=. = (1.44) - A = 'H . , A A 23 2 A =.

1) n = 1. 2 A = A A A m × 1, 7

!' # A =:

a11

a21

am1

2) m =1. 2 A = A A A 1× n , 7

A =:

A = a11 .a12 . .a1n .

3) m = n . 2 2 A = 2 2'=, A = 7 #. 2 , 2 2'=,

7 # %. 0 , 7 A =

7 - D 2. A = n - 7

3 D2 A a11, a22,..., ann , 2 4 2,

7IH 2 I II 5 A = 4 5,

7 7! # 7 ! : A =.							3 D2 A,
2 4 2,					7		' 7 ! :. !27
2 27 A =				A	23 I 7 ' 7: det A 2
A , 2 D( A) .
# &: &,	# M! % , ! * % # 7! # 7 !,
# % &!:,
		a11	0		0	0
		a11	0		0	0
		0	a22		0	0
		0	0	a33		0	,
		... ... ... ... ...
		0	0		0	ann
I 7 !.
! 23 7 A = n -					7, 23 D2 A
=,	7 ' 7 En 2
E :

	1		0	0	0
	1		0	0	0
	0		1	0	0
E =	0		0	1	0		.
	... ... ... ... ...
	0		0	0	1
4) =, D2 A 2I, 7 &! #
A = ' 7 O :		0	0		0
		0	0		0
O =		0	0		0	.
O =						.
		0	0		0

2 ' A 3 2 2'= 2 A =,

5 Omn .

' A 7 A = E , IH 7 A A

2 4 23, 2 1 D2 A 2 ', 2 7 A = O , 23 2 2 2 23 D2 A 2 '.

! 5 2 A = 2, 2:

1.% 7 A =,

2.2 4 A =,

3.A 4 A = 2,

4.A 4 A =.

	! "
!. $" XIX , '
( ,	" .

6 # # ! %

1. 6 # # .

, !. ! A = A B I 7 # %, 2 A I

A 2 IH D2 A A =

aik = bik (i =1,2,...,m; k = 1,2,...,n).

% D A 5			A = B.	(1.45)

A A, A (1.45) 23				m n 2 A
A:	a11	= b11, a12	= b12,..., amn = bmn.

2. ! *

= 7 2 4 7 A = 27 7 23 27 A =, A IH 2

2 2'=.

,!. & A + B A = A B , A IH

A, 7 A = C , 4 D2 A AA

IH D2 A A = A B , 3
cik	i = 1, 2,..., m		(1.46)
	= aik + bik	.
		k = 1,2,...n

!27 AA A = A B 23 7 '

C = A + B.

a11

a12

...

a1n

b11

b12

...

b1n

A ' A, 2 A =

a21

a22

...

a2n

;

B =

b21

b22

...

b2n

...

am1

am2

...

amm

bm1

bm2

...

bmn

	a11	+ b11	a12	+ b12	...	a1n + b1n
	a11	+ b11	a12	+ b12	...	a1n + b1n
C = A + B =	a21	+ b21	a22	+ b22	...	a2n + b2n	.
C = A + B =		...		...	...	...	.
	am1	+ bm1	am2	+ bm2	...	amn + bmn

".		2				3				5 6
	3		4	+	1		2	=	4		.
	3		4		1		2		4
	1	0	2		0	1	5		1	1 7

# # ! * (2 A 2 4 7 2).

*1.

A + B = B + A .

2. 23

( A + B) + C = A + (B + C).

3.AA 2I' A = A 2 A = O 4 A

A = A:

A + O = A.

A ' A, 2 7 A = A A 2 23 27 2 ' 2.

3. $* % !

, !. " 5# % A ! 7

A = B , 4 D2 A I D 2

IH D2 A A = A , 3

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