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Национальный минерально-сырьевой университет «Горный»

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: ( " ?, (1.34) -

? " n ! n . 5 '# 6 #' #. 0 ( n !

n ", -

. 9 xk (k = 1, 2,..., n) , -

" " " ( , - (

, ) " k -

! . 2. D = 0.

9 ! .

1) 0 ! " D1, D2,..., Dn ",

(1.29) , ) -(1.31), 9 (, ( -

2) 0 D1, D2,..., Dn ,

(1.29), " ! ! , ,

( ), -

. ! " 9 (-

' ( ( ) #

L " ', 9

# ! " " " -

! .

# n ! ! n

a x		+ a	x	+ ... + a	x	= 0,
	11 1	12	2	1n	n
a21x1 + a22x2 + ... + a2nxn = 0,
		................				(1.35)
		................				(1.35)
a	x	+ a	x	+ ... + a	x	= 0.
	n1 1	n2	2	nn n
% " , ! 9 ! aik
:	x1 = 0, x2 = 0,..., xn = 0.					: " )-

' (# / ') & '. 0 , " ) ! -

(1.35), " ! " , ", "

) '. " (, ! !

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G ( " " ) ! , -

) , " ", -

" D . 1. D 0.

0 ( ! ! -, , ?, , ,

, " " " . 2. D = 0.

0 ( ! ! ,

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!, ! . 9 ! " 9 -

( , " ( - )( " " .

# " ! " !

n ! ! n .

5 '# 1.1. (%( ' # 4 ) " # 4 " ) (

& ').

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. , 9 9 (

. !#$' # # '# n -$ " # "

r , r - " ", -

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9 .

5 '# 1.2. # n ! ! n -9 r , r - .

# & ' ( ( (

) # '" . '

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! ! ! " :

a11x1 + a12x2a21x1 + a22 x2a31x1 + a32x2

F ( 9

+ a13x3 = 0,
+ a23x3 = 0,	(1.36)
+ a33x3 = 0.

D =	a11	a12	a13
D =	a21	a22	a23	.	(1.37)
	a31	a32	a33

": 1. D 0.

# ( - : x1 = 0, x2 = 0, x3 = 0.

2. D = 0.

,, 1.1 1.2 (1.36) ,

, ) ! , 9

", ! " 1.2.

% ( D = 0 ,

( , ( , , -

, 9 , 9 ). E-

". 1) r = 2.

: , " (

! " , ".

, (, , D23 9 a23

a11

a12

(1.38)

a31

a32

9 (1.36) 9 ! -

a11x1 + a12x2 + a13 x3 = 0,

a x

+ a

x = 0.

(1.39)

31 1

C " (1.39), " !

( ! x1 x2

, D23 :

a11x1 + a12x2 = a13 x3,

+ a

= a

x .

(1.40)

31 1

F ( 9 D23 0 , (, ? -

( " x3 (" " ,

! ), :

a13x3

a12

a11

a13x3

x =

a33x3

a32

;

x =

a31

a33x3

D23

G ( " , ".

a13x3

a12

= x

a13

a12

= x

a12

a13

= x D ,

a x a

3 21

a11

a13x3

= x

a11

a13

= x D .

3 22

, ! " "

A = ( 1)2+1 D = D

; A = ( 1)2+2 D = D ;

A = ( 1)2+3 D = D

" (1.40) :

x =

A21

x , x

A22

x .

(1.41)

A23

P (1.41)

(1.39), « » .

x3 . ,

( " x3

( -

(1.41) ) " !

x2 . ,

" (1.39) -

!, " (1.41).

3#' 4#. , -

" ! ! " . 2) r = 1.

E (1.36) , ! 9


9 , ! "	9 - 9-
! (1.36)	- . , (, ,
a11 0 . 9 (1.36) 9
a11x1 + a12x2 + a13x33 = 0.		(1.42)

, 9 ( ", " (1.42) " " "

( (1.36).

E " (1.42) x1 , -

!, ( " ! "! ! x2								x3 ,
:		a12				a13
x	=		x	2			x .	(1.43)

1		a11			3
						a11

P (1.43) " (1.42). % :

( " x2 x3 (1.43) )

x1 - " " (1.42)

!, " (1.43).

3#' 4# . , -

" " " .

@ ! , !

!. @ , " ) " (, -

" . -

( ( ( ! " ! 9 ! .

! ? ?. K 9 ! " " " " " " -

9 9 . F (-

? " ( ". 9

" " " K - ( -

" !, ( -

# ) ( !

( ! - m ! n -

, ) ! !, [1].

! & .# #4

3# #4# 1. E (

2x1 3x2 + x3 = 11,x1 + 5x2 4x3 = 10,4x1 + x2 3x3 = 5.

! & . (

D =	2	3	1	= 2.
	2	3	1
	1	5	4
	4	1	3

F ( ", (,

, " ?. "

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

% ?

x =

= 3;

x =

= 1;

= 2.

3# #4# 2. G (

+ x

= 1,

3x1 + x2 + 4x3 = 3,

+ 5x

+ 2x

= 2.

! & . " ( :

		D =				1	2		1	= 0.
						1	2		1
						3		1	4
						1		5	2
% D = 0,		, . C D1.
		D1 =			1		2		1
					1		2		1
				3				1	4		0.
				2				5	2
F ( " (,												, D1 -
", , .
3.	3# #4#	E
		x	+ x				2x		= 1,
		1	2					3
		2x1 + 2x2 4x3 = 2,
		3x + 3x 6x = 5.
			1				2		3
	$( D = 0, D1 = 0, D2 = 0, D3 = 0.							# , -
( ",								"				3. ,

3x1 + 3x2 6x3 = 3 , ( , (-, .

4.	3# #4# E
	x	x + 3x				= 1,
	1		2		3		= 2,
	2x		2x	2	+ 6x		= 2,
		1		2		3
	3x		+ 4x		x		= 4.
		1	2		3

$( D = 0, D1 = 0, D2 = 0, D3 = 0. % "

! " 2, " (

! , ( -

5 3# #4# . C "

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

! &. * " " " " ! ! -

. ( 9

3 1 2

D = 2 3 4 = 31. 5 2 1

% D 0 , :

	x1 = 0;	x2 = 0;	x3 = 0.
6.	3# #4# G ( ! !
".

x	+ 3x			+ 2x	= 0,
1		2		3
2x1 2x2 + x3 = 0,
x		5x		x	= 0.
1		2		3
! &. (
	D =		1	3	2	= 0.
			1	3	2
			2	2	1
			1	5	1

% D = 0 , , ", -

! . C " . F "

, " , D11 :

D	=	2	1	= 7.

11		5	1

% D11 0 , r = 2 , (, 9-

! - (. E , -

") ( , ( ! x2 x3 :

2x2 + x3 = 2x1,

5x2 x3 = x1.

F ( 9 D11 0 , (, ? -

( x1 :

	2x1	1					2	2x1

x =	x1	1	=	3x1	, x	=	5	x1		= 8x1 .


2	D11	7			3			D11	7

% , ! (

x2 = 73 x1,

x3 = 87 x1,

x1 - ) .

@ ( -

, " (

t . 0 " (	x1	= t , ( -
t . 0 " (	7	= t , ( -
	7
			= 7t,
		x	= 7t,
		1		t .
		x2 = 3t,		t .
	x		= 8t,
		3

" #' ' 1.2

1.n ! n

2., n ! n -

( ?

3.# ? ? " -n ! n .

4." ?

5. @ ! ! ( ?

6.# ! " " !

n ! ! n .

7.J " n - "?

8.? )( " " "

, 9 m !

n ?

1.3. *# ( ' & + ' (

) #

, ( " ) -

•'#.

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•C' < '#.

•% # #" '# #.

•! & ' ( ) # '- '#.

, " ! ( "

( .

0 ( " !, ( [1],

2, . 45-75 – ! . # - " (

( /1 / 6-10.

E ( " , ) " ( -

! - . % (

( " " , )

", "! " ( !-.

# " XIX , !

" ", ! -

. $( ".

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

") " m n , " '# #.' # m × n -

" :

a11	a12	...	a1n
a21	a22	...	a2n	(1.44)
...	...	...	...	(1.44)
am1	am2	...	amn

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

J , A, B . . J, ) , " 2 ' #'. ? 9 aik

; i , k -

. @ (1.44) - ) .

E ( .

1) n = 1. 0 A m × 1, " -

% :

a11 a21

am1

2) m =1. 0 A 1× n , " -

4 :

A = a11 .a12 . .a1n .

3) m = n . 0 ,

" # #. J , , " "- ' # # '#. , " " - 9 . & n - " ( 9

a11, a22 ,..., ann , ! , " ) ! -

, " $# #$# /+ -

. # ( 9, ! , "

% 4 #$# /+. * " " A ( "

": det A A , D( A) .

? , 9 ,

, , (
	a11	0	0	0
	a11	0	0	0
	0	a22	0	0
	0	0	a33	0	,
	... ...		... ... ...
	0	0	0	ann

#$# /.

* ( " n - ", ( 9

, " 4 " En E :

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