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Ɍɟɤɫɬ ɡɚɞɚɧɢɹ ʋ ɇɚɣɬɢ ɪɚɧɝ ɦɚɬɪɢɰɵ

ȼɚɪɢɚɧɬɵ ɡɚɞɚɧɢɹ ʋ

§ 1 0 1 0 2 ·

§1 3 1 0 2 ·

4 1 2 3

1¸

3 1 1 3 1

1 4 2 7 11¹

§ 3 3 2 4 6 ·

§1 3 1 0 2·

4 1 2 3

1 3 5 3 4

1¸

2 0 4 3 6

1 4 4 7 5 ¹

§1 3 1 0 2 ·

§4 3 1 5 2 ·

2 3 4 2

4 1 2 3

1¸

3 0 3 2 1

8 2 1 8 1

§ 1 3 1 0 2 ·

§3 2 1 4 1 ·

4 1 2 3

1¸

7 1 1 7 0

§1 3 1 0 2 ·

§2 5 7 1 2 ·

3 1 0 1

4 1 2 3

1¸

4 2 1 1 1

§3 3 1 2 2 ·

§1 3 1 0 2 ·

4 1 2 3

2 2 2 4

1¸

7 4 3 5 1

3 5 1 4

§2 3 3 0 4 ·

§ 1 3 1 0 2 ·

4 1 2 3

3 4 1 0

1¸

6 4 5 3 3

2 7 0 0 1

§1 3 1 0 2 ·

§1 2 5 4 1 ·

2 4 7 3

4 1 2 3

1¸

5 3 3 7 0

3 7 6 3 1 ¹

§3 2 3 10 0·

§1 2 1 3 4 ·

2 4 5 3 1

4 1 2 3

1¸

1 2 2 1 0

5 3 1 0 3

§3 2 1 5 2·

§2 3 1 5 1 ·

4 5 6 1 0

6 1 0 2 0 ¸

2 3 1 2 4

4 2 1 3

1¹

§5 3 2 4 4·

§1 3 4 5 0 ·

1 3 2 5 7

3 4 2 1

2¸

2 1 3 4 6

2 1 2 4 2

§2 1 1 5 2·

§4 3 2 0 1·

4 5 3 2 1

2 0 1 5

2 3 0 1 0

2 5 2 1 6

§2 4 6 3 1·

§ 3 4 0 1 2·

2 2 0 1 0

2 0 1 3 5

3 4 1 2 6

1 4 1 4 7

§ 1 2 3 1 2·

§ 3 1 1 4 3·

2 1 0 1 3

¨ 1 3 4 3 2

1 0 1 3 0

§2 1 3 2 4·

§1 3 1 0 2 ·

4 2 5 1 7

4 1 2 3 1¸

2 1 1 8 2

5 4 1 3 1

3.6 ǸȏȘșȌȓȢ ȒȏȔȌȐȔȢȜ ȚȗȇȉȔȌȔȏȐ

Ɉɩɪɟɞɟɥɟɧɢɟ 6 ɋɢɫɬɟɦɨɣ m ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɫ ɧɚɡɵɜɚɟɬɫɹ ɫɢɫɬɟɦɚ ɜɢɞɚ

¦ a x		a x	..... a x b
£	11 1	12 2	1n n	1
¦	11 1	12 2	1n n	1
¦		a x	a x	b
¦ a x		a x	a x	b
¦	21 1	22 2	2n n	2	.
¤					.
¦ ......................................
¦
¦
¦			a x	b
¦a x a x			a x	b
¥	m1 1	m2 2	mn n	m

n ɧɟɢɡɜɟɫɬɧɵɦɢ

3.6.1)

Ɂɞɟɫɶ aij ,bj i 1, ..., n, j 1, ..., m ɜɟɳɟɫɬɜɟɧɧɵɟɱɢɫɥɚ ɧɚɡɵɜɚɟɦɵɟɤɨɷɮ ɮɢɰɢɟɧɬɚɦɢ ɫɢɫɬɟɦɵ x j j 1, ..., m – ɩɟɪɟɦɟɧɧɵɟ

ȿɫɥɢ ɜɫɟ ɤɨɷɮɮɢɰɢɟɧɬɵ bj 0 j 1, ..., m ɬɨ ɫɢɫɬɟɦɚ ɧɚɡɵɜɚɟɬɫɹ ɨɞɧɨ ɪɨɞɧɨɣ ȿɫɥɢ ɫɭɳɟɫɬɜɭɟɬ ɯɨɬɹ ɛɵ ɨɞɢɧ bj v0, ɬɨ ɫɢɫɬɟɦɚ ɧɚɡɵɜɚɟɬɫɹ ɧɟɨɞɧɨ

ɪɨɞɧɨɣ

ɇɚɛɨɪ ɱɢɫɟɥ B1, B2, ..., Bn ɧɚɡɵɜɚɟɬɫɹ ɪɟɲɟɧɢɟɦ ɫɢɫɬɟɦɵ 3.6 ɟɫɥɢ ɩɪɢ

ɩɨɞɫɬɚɧɨɜɤɟ ɢɯ ɜ ɫɢɫɬɟɦɭ ɜɦɟɫɬɨ ɩɟɪɟɦɟɧɧɵɯ ɜɫɟ ɭɪɚɜɧɟɧɢɹ ɫɢɫɬɟɦɵ ɩɪɟɜɪɚ ɳɚɸɬɫɹ ɜ ɬɨɠɞɟɫɬɜɚ ɋɢɫɬɟɦɚ ɢɦɟɸɳɚɹ ɯɨɬɹ ɛɵ ɨɞɧɨ ɪɟɲɟɧɢɟ ɧɚɡɵɜɚɟɬɫɹ ɫɨɜɦɟɫɬɧɨɣ ȿɫɥɢ ɫɢɫɬɟɦɚ ɢɦɟɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɪɟɲɟɧɢɟ ɬɨ ɨɧɚ ɧɚɡɵɜɚɟɬɫɹ ɨɩɪɟɞɟɥɟɧɧɨɣ

ɉɪɢɦɟɪ6 ɋɢɫɬɟɦɚ

£¦x 2y 1

¦¦¥x 2y 3

ɧɟ ɢɦɟɟɬ ɪɟɲɟɧɢɣ ɗɬɨ ɧɟɫɨɜɦɟɫɬɧɚɹ ɫɢɫɬɟɦɚ

ɋɢɫɬɟɦɚ

£¦ x y 2

¦¦¥ 2x 2y 4

ɢɦɟɟɬ ɛɟɫɤɨɧɟɱɧɨɟ ɤɨɥɢɱɟɫɬɜɨ ɪɟɲɟɧɢɣ ɗɬɚ ɫɢɫɬɟɦɚ ɫɨɜɦɟɫɬɧɚɹ ɧɨ ɧɟ ɨɩɪɟ ɞɟɥɟɧɧɚɹ

ɋɢɫɬɟɦɚ

£¦x y 3

¦¦¥x y 7

ɢɦɟɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɪɟɲɟɧɢɟ

£¦x 5 ¦ .

¦¦¥y 2

ɗɬɚ ɫɢɫɬɟɦɚ ɫɨɜɦɟɫɬɧɚɹ ɢ ɨɩɪɟɞɟɥɟɧɧɚɹ Ɇɚɬɪɢɰɚ

a11

Aa21

....

am1

a	.......	a	¬
12		1n

a	.......	a
a	.......	a
22		2n

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


am2	.......
am2	.......	amn®

ɧɚɡɵɜɚɟɬɫɹ ɦɚɬɪɢɰɟɣ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɫɢɫɬɟɦɵ ɫɬɨɥɛɟɰ

b1 ¬

b b2

...

bm®

ɹɜɥɹɟɬɫɹ ɫɬɨɥɛɰɨɦ ɫɜɨɛɨɞɧɵɯ ɱɥɟɧɨɜ ɫɢɫɬɟɦɵ ɋɬɨɥɛɟɰ

x1 ¬

x x2

...

xn®

ɹɜɥɹɟɬɫɹ ɫɬɨɥɛɰɨɦ ɩɟɪɟɦɟɧɧɵɯ ɋ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɷɬɢɯ ɨɛɨɡɧɚɱɟɧɢɣ ɫɢɫɬɟ ɦɭ 3.6 ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɜ ɦɚɬɪɢɱɧɨɣ ɮɨɪɦɟ

Ax b.

3.6.2)

Ɇɚɬɪɢɰɭ A b ɤ ɦɚɬɪɢɰɟ ɤɨɷɮɮɢɰɢɟɧɬɨɜ A ɫɩɪɚɜɚ ɩɪɢɩɢɫɚɧ ɫɬɨɥɛɟɰ ɫɜɨɛɨɞɧɵɯ ɱɥɟɧɨɜ b ɧɚɡɵɜɚɸɬ ɪɚɫɲɢɪɟɧɧɨɣ ɦɚɬɪɢɰɟɣ ɫɢɫɬɟɦɵ ɋɢɫɬɟ ɦɚ 3.6 ɜ ɦɚɬɪɢɱɧɨɣ ɡɚɩɢɫɢ 3.6 ɩɨɥɧɨɫɬɶɸ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɜɨɟɣ ɪɚɫɲɢ ɪɟɧɧɨɣ ɦɚɬɪɢɰɟɣ

ȼɦɟɫɬɟ ɫ ɫɢɫɬɟɦɨɣ 3.6 ɪɚɫɫɦɨɬɪɢɦ ɨɞɧɨɪɨɞɧɭɸ ɫɢɫɬɟɦɭ

Ax 0.

3.6.3)

ǸȉȕȐȘșȉȇ ȗȌȟȌȔȏȐ ȒȏȔȌȐȔȢȜ ȘȏȘșȌȓ [5]

1.ɋɭɦɦɚ ɞɜɭɯ ɪɟɲɟɧɢɣ ɨɞɧɨɪɨɞɧɨɣ ɫɢɫɬɟɦɵ ɟɫɬɶ ɪɟɲɟɧɢɟ ɫɢ ɫɬɟɦɵ

2.Ɋɚɡɧɨɫɬɶ ɞɜɭɯ ɪɟɲɟɧɢɣ ɧɟɨɞɧɨɪɨɞɧɨɣ ɫɢɫɬɟɦɵ ɟɫɬɶ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɣ ɫɢɫɬɟɦɵ

3.ɋɭɦɦɚ ɪɟɲɟɧɢɹ ɧɟɨɞɧɨɪɨɞɧɨɣ ɫɢɫɬɟɦɵ ɢ ɪɟɲɟɧɢɹ ɨɞɧɨɪɨɞ ɧɨɣ ɫɢɫɬɟɦɵ ɟɫɬɶ ɪɟɲɟɧɢɟ ɧɟɨɞɧɨɪɨɞɧɨɣ ɫɢɫɬɟɦɵ 6.2).

ǳȌșȕȋ ǱȗȇȓȌȗȇ ȗȌȟȌȔȏȦ ȒȏȔȌȐȔȢȜ ȘȏȘșȌȓ

Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜ ɤɨɬɨɪɨɣ ɤɨɥɢɱɟɫɬɜɨ ɭɪɚɜɧɟ ɧɢɣ ɢ ɤɨɥɢɱɟɫɬɜɨ ɩɟɪɟɦɟɧɧɵɯ ɫɨɜɩɚɞɚɟɬ

¦ a x a

..... a

11 1

..... a

¦a

¦ 21 1

.4)

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

..... a

¦a

n1 1

ɉɭɫɬɶ

.......

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

.....

an2 .......

an1

ann ®

ɹɜɥɹɟɬɫɹ ɦɚɬɪɢɰɟɣ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɫɢɫɬɟɦɵ.

Ɍɟɨɪɟɦɚ[5, 11, 12]. ȿɫɥɢ det A v0 ɬɨ ɫɢɫɬɟɦɚ 3.6.4 ɢɦɟɟɬ ɟɞɢɧ

ɫɬɜɟɧɧɨɟ ɪɟɲɟɧɢɟ ɤɨɬɨɪɨɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɭɥɚɦɢ x

, i 1, ..., n ɝɞɟ

% det A , %i – ɨɩɪɟɞɟɥɢɬɟɥɶ ɦɚɬɪɢɰɵ ɤɨɬɨɪɚɹ ɩɨɥɭɱɢɬɫɹ ɟɫɥɢ ɜ ɦɚɬɪɢɰɟ A ɫɬɨɥɛɟɰ ɫ ɧɨɦɟɪɨɦ i ɡɚɦɟɧɢɬɶ ɫɬɨɥɛɰɨɦ ɫɜɨɛɨɞɧɵɯ ɱɥɟɧɨɜ b.

ɉɪɢɦɟɪ Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ

£¦2x 3y 4 ¦ .

¦¦¥ 3x 2y 7

%	2	3	13, %x	4	3	13, %y	2	4	26.
	3	2		7	2		3	7

ɉɨɥɭɱɚɟɦ

x 13 1, y 26 2.13 13

Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ

									¦x 2y 3z 14
							£
							¦
							¦
									¤x y 2z 7 .
							¦
							¦						2x z 5
							¦						2x z 5
							¥¦
				1	2		3				5					2		3			5 2


%				1 1			2				5 1							2						5,
				2	0		1						0			0		1			5		1
				2	0		1						0			0		1
		14			2		3							1		2		3				1 2


%x		7 1 2												3 1 2											5,
		5			0		1							0		0		1				3	1
		5			0		1							0		0		1
			1		14		3					5 1						3			5		1


%y			1		7		2					5				3		2						10.
			2		5		1						0			0		1			5		3
			2		5		1						0			0		1
	1				2		14			1			2					14	3				21


%z	1				1		7			0			3			21								15.
	2 0						5			0 4 23									4 23
	2 0						5			0 4 23
ɉɨɥɭɱɚɟɦ
				x		5		1, y							10		2, z			15			3.
				x				1, y									2, z						3.
					5								5								5

Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ

£¦2x 2y z 4

¦¤ x 3y z 6 .

¦ x y z 4

¦¥

		2		2		1		1		3	0	1		3
		2		2		1		1		3	0
%		1			3	1		2		4	0				10,
		1			1	1		0		0	1	2		4
		1			1	1		0		0	1
	4			2		1		0		3 0			0	3


%x	6				3	1		10		4	0					30,
	4				1	1		4		1	1		10	4
	4				1	1		4		1	1
				2	4	1			1	0	0	6 1


%y				1 6		1			1 6		1				10
				1	4	1			1	4	1	4		1
				1	4	1			1	4	1
			2		2	4	0			4	4		4 4


%z			1 3 6				0 2				2					0.
			1		1	4	1			1	4		2		2
			1		1	4	1			1	4

ɉɨɥɭɱɚɟɦ

x 1030 3, y 1010 1, z 100 0.

Ɍɟɤɫɬ ɡɚɞɚɧɢɹ ʋ 6 Ɋɟɲɢɬɶ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ Ax b ɦɟ ɬɨɞɨɦ Ʉɪɚɦɟɪɚ.

ȼɚɪɢɚɧɬɵ ɡɚɞɚɧɢɹ ʋ

§ 1 2 3·

§ 14 ·

§ 1 2

4 ·

§ 23 ·

2 3 5

2 5

§ 1 2

2 ·

§ 7 ·

§ 1 2 1·

§ 1·

2 5

1 3 2

3 5 6

3 5 3

§ 1 2

3 ·

§ 11 ·

§ 1 2 2 ·

§ 5 ·

1 3

1 3 1

3 5 6

§ 1 2 3·

§ 5 ·

§ 1 2 1 ·

§ 1 ·

1 3

1 3 0

3 5 3

§ 1 2 2·

§ 1 ·

§ 1 2 3 ·

§ 8 ·

1 3 3

0 1 1

3 5 6

§ 1 2 1·

§ 1 ·

§ 1 2

4 ·

§ 15 ·

1 1 0

0 1 1

3 5 3

§ 1 2 3 ·

§ 14 ·

§ 1 2 0 ·

§ 1·

2 5 5

2 5 1¸

3 5 0

§ 1 2 3·

§ 2 ·

§ 1 2

4 ·

§ ·

2 5 7

1 3

§ 1 2 2·

§ 1·

§ 1 2 0 ·

§ 1·

2 5 5

1 3 1¸

3 5 6

3 5 0

§ 1 2 1·

§ 2·

§ 1 2 1 ·

§ 2 ·

2 5 3

2 5 1

3 5 3

§ 1 2 3·

§ 11 ·

§ 1 2 2·

§ 5 ·

1 1 2

1 1 1

3 5 6

§ 1 2

4 ·

§ 7 ·

§ 1 2 1 ·

§ 0 ·

0 1 1¸

3 5 3

3 5 12¹

§ 1 2 2 ·

§ 1·

§ 1 2 3 ·

§ 2 ·

2 3 5

2 3 7

3 5 6

§ 1 2 0 ·

§ 1·

§ 1 2 1 ·

§ 2·

2 3 3

2 3 1¸

3 5 0

3 5 3

§ 1 2 2·

§ 7 ·

§ 1 2 1·

§ 2 ·

2 3 3

2 3 1

3 5 6

3 5 3

Ɋɚɫɫɦɨɬɪɢɦ ɨɩɹɬɶ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɨɛɳɟɝɨ ɜɢɞɚ

¦ a x		a x	..... a x b
£	11 1	12 2	1n n	1
¦	11 1	12 2	1n n	1
¦		a x	a x	b
¦ a x		a x	a x	b
¦	21 1	22 2	2n n	2	.	3.6.5)
¤	......................................				.	3.6.5)
¦	......................................
¦
¦
¦
¦a x a x ..... a x b
¥	m1 1	m2 2	mn n	m

Ɍɟɨɪɟɦɚ Ʉɪɨɧɟɤɟɪɚ – Ʉɚɩɟɥɥɢ [5, 11, 12]. Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɫɢɫɬɟ ɦɚ 3.6 ɢɦɟɥɚ ɪɟɲɟɧɢɟ ɧɟɨɛɯɨɞɢɦɨ ɢ ɞɨɫɬɚɬɨɱɧɨ ɱɬɨɛɵ ɪɚɧɝ ɦɚɬɪɢɰɵ ɫɢ ɫɬɟɦɵ ɛɵɥ ɪɚɜɟɧ ɪɚɧɝɭ ɪɚɫɲɢɪɟɧɧɨɣ ɦɚɬɪɢɰɵ

rangA rang A b

3.6.6)

ǳȌșȕȋ ǪȇȚȘȘȇ ȗȌȟȌȔȏȦ ȒȏȔȌȐȔȢȜ ȘȏȘșȌȓ

ɫɨɫɬɨɢɬ ɜ ɬɨɦ ɱɬɨɛɵ ɫ ɩɨɦɨɳɶɸ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɧɟ ɢɡɦɟɧɹɸɳɢɯ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɩɪɢɜɟɫɬɢ ɪɚɫɲɢɪɟɧɧɭɸ ɦɚɬɪɢɰɭ ɤ ɬɚɤɨɦɭ ɜɢɞɭ ɤɨɝɞɚ ɧɚ ɦɟɫɬɟ ɛɚ ɡɢɫɧɨɝɨ ɦɢɧɨɪɚ ɫɬɨɢɬ ɟɞɢɧɢɱɧɚɹ ɦɚɬɪɢɰɚ

ɉɭɫɬɶ ɛɚɡɢɫɧɵɣ ɦɢɧɨɪ ɪɚɫɩɨɥɨɠɟɧ ɜ ɩɟɪɜɵɯ r ɫɬɪɨɤɚɯ ɢ r ɫɬɨɥɛɰɚɯ ɦɚɬ ɪɢɰɵ Ɍɨɝɞɚ ɩɨɫɥɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɪɚɫɲɢɪɟɧɧɚɹ ɦɚɬɪɢɰɚ ɩɪɢɧɢɦɚɟɬ ɜɢɞ

	1 .....	0
	1 .....	0

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


	0 ......	1
	0 ......	1


	0 ......	0
	0 ......	0


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


	0 ......	0
	0 ......	0

a	.....	a	B		¬
1r 1		1n	1

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


a	.....	a	B
a	.....	a	B	r
rr 1		rn		r		3.6.7)
					.	3.6.7)
0	.....	0	0
0	.....	0	0


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



0	.....	0	0
0	.....	0	0		®

ȼɨɡɜɪɚɳɚɹɫɶ ɨɬ ɦɚɬɪɢɰɵ 3.6.7 ɤ ɫɢɫɬɟɦɟ ɩɨɥɭɱɚɟɦ

¦ x		a	x	...... a	x	B
£	1	1r 1 r 1		1n n		1
¦
¦		a	x	...... a	x	B
¦x							2
¦	2	2r 1	r 1	2n n				.
¤		.............................
¦
¦
¦		a	x	...... a	x	B
¦ x							r
¥¦	r	rr 1	r 1	rn n

ɉɟɪɟɦɟɧɧɵɟ x1, x2, .., xr ɧɚɡɵɜɚɸɬɫɹ ɛɚɡɢɫɧɵɦɢ ɩɟɪɟɦɟɧɧɵɦɢ ɉɟɪɟɦɟɧ ɧɵɟ xr 1, xr 2, .., xn – ɩɚɪɚɦɟɬɪɢɱɟɫɤɢɟ ɉɪɢ ɥɸɛɵɯ ɡɧɚɱɟɧɢɹɯ ɩɚɪɚɦɟɬɪɨɜ Cr 1, Cr 2, ..., Cn ɦɵ ɩɨɥɭɱɢɦ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ 3.6.5):

¦ x B

1r 1 r 1

1n n

¦x

2r 1 r 1

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

¤ xr

arr 1xr 1

arnxn

¦ x

r 1

¦ ....................................................

¦ x

¥¦

ȄȒȌȓȌȔșȇȗȔȢȌ ȖȗȌȕȈȗȇȎȕȉȇȔȏȦ ȓȇșȗȏȝȢ

1.ɉɟɪɟɫɬɚɧɨɜɤɚ ɫɬɪɨɤ ɦɚɬɪɢɰɵ

2.ɍɦɧɨɠɟɧɢɟ ɫɬɪɨɤɢ ɧɚ ɱɢɫɥɨ ɨɬɥɢɱɧɨɟ ɨɬ ɧɭɥɹ

3.Ⱦɨɛɚɜɥɟɧɢɟ ɤ ɨɞɧɨɣ ɫɬɪɨɤɟ ɞɪɭɝɨɣ ɫɬɪɨɤɢ ɭɦɧɨɠɟɧɧɨɣ ɧɚ ɱɢɫɥɨ Ɍɟɨɪɟɦɚɗɥɟɦɟɧɬɚɪɧɵɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɪɚɫɲɢɪɟɧɧɨɣ ɦɚɬɪɢɰɵ ɧɟ

ɢɡɦɟɧɹɸɬ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɬɟɨɪɟɦɵ ɨɱɟɜɢɞɧɵɦ ɨɛɪɚɡɨɦ ɜɵɬɟɤɚɟɬ ɢɡ ɬɨɝɨ ɱɬɨ ɞɟɣ

ɫɬɜɢɹ ɧɚɞ ɫɬɪɨɤɚɦɢ ɪɚɫɲɢɪɟɧɧɨɣ ɦɚɬɪɢɰɵ ɪɚɜɧɨɫɢɥɶɧɵ ɞɟɣɫɬɜɢɹɦ ɧɚɞ ɭɪɚɜ ɧɟɧɢɹɦɢ ɫɢɫɬɟɦɵ

ɉɪɢɦɟɪ Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ

£¦ 3x y z 0

¦¤x 3y 2z 6.

¦ 2x 4y z 8

¦¥

Ɂɚɩɢɲɟɦ ɪɚɫɲɢɪɟɧɧɭɸ ɦɚɬɪɢɰɭ ɫɢɫɬɟɦɵ
3 1 1		0		¬ 1 3 2						6¬ 1						3 2		6¬
3 1 1		0		¬ 1 3 2						6¬ 1						3 2		6¬


								1		0						8	5	18
1 3 2		6				3 1		1		0				0		8	5	18


	4 1	8								8								20
2	4 1	8		® 2 4 1						8		® 0 10 5						20		®

1 3 2			6¬ 1 3 2								6¬ 1 3 2								6¬
1 3 2			6¬ 1 3 2								6¬ 1 3 2								6¬


	2 1		4								4						1/ 2
0	2 1		4			0 2 1					4			0 1			1/ 2		2


	8 5		18						1		2						1		2
0	8 5		18		® 0 0				1		2		® 0 0				1		2		®

				1 3 0			2¬ 1 0 0								1	¬
				1 3 0			2¬ 1 0 0								1	¬



				0 1 0			1		0 1 0						1.


							2								2
				0 0 1			2	®	0 0 1						2	®

Ɋɟɲɟɧɢɟ ɫɢɫɬɟɦɵ

x 1, y 1, z 2.

Ɂɚɦɟɱɚɧɢɟ Ʉɚɤ ɜɢɞɧɨ ɢɡ ɩɪɢɦɟɪɚ ɜɵɲɟ ɟɫɥɢ ɫɢɫɬɟɦɚ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟ ɧɢɣ ɢɦɟɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɪɟɲɟɧɢɟ ɬɨ ɷɬɨ ɪɟɲɟɧɢɟ ɛɭɞɟɬ ɡɚɩɢɫɚɧɨ ɩɨɫɥɟ ɩɪɟ ɨɛɪɚɡɨɜɚɧɢɣ ɦɟɬɨɞɚ Ƚɚɭɫɫɚ ɜ ɩɨɫɥɟɞɧɟɦ ɫɬɨɥɛɰɟ ɪɚɫɲɢɪɟɧɧɨɣ ɦɚɬɪɢɰɵ

Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ

					¦ x 2y 4z 1
					£
					¦
					¦				2.
					¤x 3y 2z				2.
					¦
					¦
					¦x 4y 8z 3
					¥¦
Ɂɚɩɢɲɟɦ ɪɚɫɲɢɪɟɧɧɭɸ ɦɚɬɪɢɰɭ ɫɢɫɬɟɦɵ
1 2 4				1 ¬		1 3 2				2¬ 1 3 2			2¬
1 2 4				1 ¬		1 3 2				2¬ 1 3 2			2¬


	1	3 2
	1	3 2		2	1 2 4					1	0 1 6		1


	1	4 8				1 4 8
	1	4 8		3		1 4 8				3	0 1 6		1
				®						®			®
			1 3 2				2¬ 1 0 16					1¬
			1 3 2				2¬ 1 0 16					1¬


						6	1				6
			0 1			6	1		0 1		6	1.


						0	0				0
			0 0			0	0		0 0		0	0
								®				®

Ȼɚɡɢɫɧɵɣɦɢɧɨɪɪɚɫɩɨɥɨɠɟɧɜɩɟɪɜɨɦɢɜɬɨɪɨɦɫɬɨɥɛɰɚɯ ɋɥɟɞɨɜɚɬɟɥɶɧɨ x, y – ɛɚɡɢɫɧɵɟ ɩɟɪɟɦɟɧɧɵɟ, z – ɩɚɪɚɦɟɬɪɢɱɟɫɤɚɹ ɩɟɪɟɦɟɧɧɚɹ ɉɨɥɚɝɚɹ z C

ɡɞɟɫɶ C – ɩɪɨɢɡɜɨɥɶɧɨɟ ɜɟɳɟɫɬɜɟɧɧɨɟ ɱɢɫɥɨ , ɩɨɥɭɱɚɟɦ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ x 1 16C, y 1 6C, z C C R ).

Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ

¦ 2x 3y z t 1
£
¦
¦	x 2y t 1
¦	x 2y t 1
¦		.
¤		.
¦4x 7 y z 3t 3
¦
¦
¦	3y z 2t 2
¦ x	3y z 2t 2
¥

Ɂɚɩɢɲɟɦ ɪɚɫɲɢɪɟɧɧɭɸ ɦɚɬɪɢɰɭ ɫɢɫɬɟɦɵ:
2 3 1 1		1¬	1 2 0 1		1¬
2 3 1 1		1¬	1 2 0 1		1¬



1 2 0 1		1	2 3 1 1		1


	1 3			1 3
4 7	1 3	3	4 7	1 3	3



1 3 1 2		2®	1 3 1 2		2®

1 2		0	1	1¬	1 2 0 1	1¬		1 0 2 1		1	¬


		1	1	1					1
0 1		1	1	1	0 1 1 1	1		0 1 1	1	1


	1	1	1	1		0			0	0	.
0	1	1	1	1	0 0 0 0	0		0 0 0	0	0


		1	1			0			0	0
0 1		1	1	1®	0 0 0 0	0	®	0 0 0	0	0	®

					100

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