Методичка

											11
23.		4	−1	3							24.		x − 5	7	5
		4 x + 3		1					> 0				8	x + 3 8			< 0
		− 8	2	x + 5									1	0	1
25.		9	7	1							26.		x −10	2	9
		0	x −1	8		> 0							− 1	x + 3 1								< 0
		9	7	x + 5									5	− 5	− 5
27.		3	2	7							28.		x + 7	4	1
		− 3 x + 1					9			> 0			2	x −1 − 2					< 0
		6	4	x + 16									9	0	− 9
29.		3	− 1	3							30.		x + 8	5	2
		0	x + 8	7			> 0						2	x − 3 − 2						< 0
		6	− 2	x − 9									− 1	0	1

1.2.3. Обчислити визначник 4-го порядку.

а) розклавши по елементам 1-го рядка; б) використавши властивості визначників.

1.		2	− 1	1	0		2.		1	2	3	4		3.		0	1	1	1
		0	1	2	−1				2	3	4	1				1	0	1	1
		3	− 1	2	3				− 3 4		1	2				2	2	1	1
		3	1	6	1				4	1	2	3				1	1	1	0

4.		1	2	3	4		5.		1	1	1	1		6.		2	1	0	2
		− 2	1	5	6				1	−1	1	1				3	2	1	0
		− 3	− 5	1	7				1	1	−1 1					− 1	0	1	3
		− 4	− 6	− 7 1					1	−1	1	− 1				− 1	2	1	3

7.		1	2	2	3		8.		1	2	2	2		9.		−1	4	4	4
		3	−1 − 4		− 6				− 2	2	2	2				− 4	0	5	4
		1	0	0	1				− 2	− 2	3	2				− 2	− 2	3	2
		1	2	−1	− 7				− 2	− 2	− 2 4					− 4	− 4	1	6

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12

10.		3	5			7					2		11.				6	4	0		7										12.					1	2		3		0
		1	2			3					4						3	4	4		7															3	1		3		5
		− 2 − 3				3 2											0	2 4 8																		5	3		1		3
		1	3			5					4						3	1	2		5															2	3		0		1
13.		1	3	5		3						14.					8		5		7			2							15.					3	1		− 2					1
		0	1	2		3											3		2		3 4															5	2		− 3 3
		3	0	1	2												− 5 − 3				3 2															7	3		3 5
		5	3	1		3											4		3		5			4												2	4		2					4
16.		3	− 1		2	3						17.					1	1	1					1							18.					1	2		5		2
		3	1		6	1											1	0	0					2												0	1		3		4
		2	0 3 −1														1 1 −1 1																			3	0		1		2
		2	− 1 1 0														1 1 1 −1																			2	3		3		4
19.		1	3	5		3						20.			1			4		5				3								21.		3				5		7				2
		0	1	2		3									3 − 5 −10 − 6																			1				2 3 4
		3	0	1	2										1 0					1				1										− 2 − 3 3 2
		5	3	1		3									1			1		6				7										1				3		5				4
22.		2	0	1		12						23.					1	3	5		2										24.					3		2		1				0
		3	1	2 14													2	1 3 3																		− 2 2 2 6
		0	2	3 20													3	3 1 0																		−1 2 1 3
		1	3	0		14											0	5	3		1															2		1		0				2
25.		1	1	− 1		1						26.					3		2		3			4							27.					1	1	2		1
		2	0	0		2											4		3		5			4												2	1	2		1
		3	1	1		3											− 2 −1			6 6																1	1 0 1
		1	1	1 −1													8		5	7				2												1	1 0 1

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										13
28.		3	1	0	2		29.		3	2	2	2		30.		1	3	0	1
		5	2	1	0				0	2	2	2				2	5	-1	3
		-1	0	1	3				- 4	2	3	2				3	7	6	5
		-1	2	1	3				- 4	- 2 - 2 4						4	2	6	4

1.2.4 Розв′язати систему рівнянь :

а) за правилом Крамера ; б)матричним способом; в) методом Гауса.

ì 2x1 - 3x2 + x3 = 0

1.ïí-2x1 - 3x2 + 4x3 = -1 ïî x1 - x2 + = 33x3

ì2x1 + 3x2 + x3 = 5

3.ïí 3x1 + 2x2 + x3 = 4 ïî4x1 + 2x2 - 3x3 = -4

ìx1 - 5x2 + 2x3 = -11

5.ïí -x1 + 2x2 - x3 = 5 ïî 2x1 - x2 + 3x3 = -4

	ì2x + 3x + 2x = 4
7.	ï	1	2	3
	í	x1 + 2x2 - x3 = 5
	ï	-x	+ x	= -1
	î	1	3

ì 3x1 - x2 + x3 = 4

9.ïí2x1 - 5x2 - 3x3 = -17 ï x = 0

î 1 + x2 - x3

ì2x1 + 4x2 + 3x3 =1 11. ïí x1 - 2x2 + 4x3 = 3 ïî 3x1 - x2 + 5x3 = 2

	ìx - 2x - x = -2
2.	ï	1	2		3
	í	2x1 + x2 + 3x3 =1
	ï-x - 3x				+ x	= 2
	î	1		2	3
	ì	x - x		+ 3x		= -4
4.	ï	1	2		3
	í	-x1 + 2x2 + x3 = 2
	ï
	î3x1 + 5x2 + 2x3 =11
	ì 3x1 + x2 + x3 = 4
6.	ï-3x + 2x - 2x = - 1
	í	1			2	3
	ï	-x - 3x + 4x = -4
	î	1		2		3
	ì6x + x - 5x = -17
8.	ï	1	2		3
	í	x1 + x2 + x3 = 0
	ï	-2x - 3x = -4
	î		1		3

ì x1 + x2 + x3 = 2

10.ïí2x1 - x2 - 6x3 = -1 ï 3x - 2x = 8

î 1 2

ì 2x1 - 3x2 + x3 = 2

12.ïíx1 + 5x2 - 4x3 = -5 ïî4x1 + x2 - 3x3 = -4

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ì2x1 - 4x2 + 3x3 =1 13. ïí x1 - 2x2 + 4x3 = 3 ïî 3x1 - x2 + 5x3 = 2

	ìx1 + 2x2 + 3x3 = 5
15.	ï	2x - x				- x		=1
	í	1	2				3
	ïx + 3x		2		+ 4x			= 6
	î	1					3
	ì4x + 3x - 2x = 4
17.	ï	1		2				3
	í	x1 + 2x2 + x3 = 8
	ï3x + 2x				2	+ x		=10
	î	1					3
	ì x1 - x2 + x3 = -2
19.	ï	-2x + x			2	- 3x =1
	í	1						3
	ï					+ 2x3 = 3
	î3x1 + 2x2
	ì x1 + x2 + 2x3 = 3
21.	ï	-2x + x					- 3x =1
	í	1			2			3
	ï					+ 5x3 = -2
	î3x1 - 2x2
23.	ì 2x1 - 3x2 + 4x3 = 3
	ï	x1 + x2				+ x3 = 2
	í
	ï					+ 2x3 = -1
	î3x1 - 2x2
	ì2x + 3x - 3x = -5
25.	ï	1		2				3
	í	x1 + x2 + x3 = 2
	ï	3x - x				+ 2x = 0
	î	1		2				3

ìx1 - 2x2 - x3 = -2

27.ïí 2x1 + x2 + 3x3 =1 ïî-x1 + 3x2 + x3 = 2

14

ì 2x1 - x2 + x3 = 2

14.ïí3x1 + 2x2 + 2x3 = -2 ïî x1 - =12x2 + x3

	ìx + 2x			+ 3x =14
16.	ï	1	2	3
	í	x1 + x2 + x3 = 6
	ï		x1 + x2 = 3
	î

ì4x1 + 3x2 + 2x3 =16

18. ïí x1 + 2x2 + 3x3 =14 ïî 5x1 - x2 - x3 = 0

	ì x1 + x2 - 2x3 = -1
20.	ï	3x + x			+ 4x				= 3
	í	1		2			3
	ïx - 2x + 3x = -2
	î	1	2				3
	ì 2x1 - x2 + x3 = -1
22.	ï x - 3x			2	+ 2x				= 0
	í	1					3
	ïx + 2x - 2x = -3
	î	1	2				3
	ì3x1 + 2x2 + x3 =1
24.	ïx - 3x + 2x							= 0
	í	1	2				3
	ï-2x		+ x			+ x		= 5
	î	1		2			3
	ì 2x1 - 3x2 + x3 = 0
26.	ï-2x		- 3x				+ 4x = -1
	í	1			2				3
	ï	x - x				+ 3x			= 3
	î	1		2			3

ì 2x1 + 3x2 + x3 = 5

28.ïí 3x1 + 2x2 + x3 = 4 ïî4x1 + 2x2 - 3x3 = -4

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15

	ì 2x1 - x2 + x3 = 2				ì 5x1 + 8x2 - x3 = 7
29.	ïx - x - 3x = -3			30.	ï x + 2x		+ 3x =1
	í 1	2	3		í 1	2	3
	ï	+ x2	- 4x3 = 2		ï	- 3x2	+ 2x3 = 9
	î5x1				î2x1

1.2.5. Знайти розв′язки однорідних систем лінійних рівнянь.

ì2x1 - x2 + 2x3 = 0 1.а) ïí x1 + x2 + 2x3 = 0 ïî4x1 + x2 + 4x3 = 0

ì3x1 - x2 + 2x3 = 0 2. .а) ïí x1 + x2 + x3 = 0 ïîx1 + 3x2 + 3x3 = 0

ì x1 + 3x2 + 2x3 = 0

3. .а) ïí 2x1 - x2 + 3x3 = 0

ïî3x1 - 5x2 + 4x3 = 0

ì4x1 - x2 +10x3 = 0

4. .а) ïí x1 + 2x2 - x3 = 0 ïî2x1 - 3x2 + 4x3 = 0

ì2x1 + 5x2 + x3 = 0

5. .а) ïí4x1 + 6x2 + 3x3 = 0 ïî x1 - x2 - 2x3 = 0

ì3x1 - x2 - 3x3 = 0 6. .а) ïí2x1 + 3x2 + x3 = 0 ïî x1 + x2 + 3x3 = 0

б)

б)

б)

б)

б)

б)

ì x1 + x2 + x3 = 0 ïí 2x1 - 3x2 + 4x3 = 0

ïî4x1 -11x2 +10x3 = 0

ì5x1 - 6x2 + 4x3 = 0 ïí 3x1 - 3x2 + x3 = 0 ïî2x1 - 3x2 + 3x3 = 0

ìx1 + 2x2 - 5x3 = 0

ïí 2x1 - 4x2 + x3 = 0 ïî3x1 - 2x2 - 4x3 = 0

ìx1 + x2 + x3 = 0

ïí2x1 - 3x2 + 4x3 = 0 ïî3x1 - 2x2 + 5x3 = 0

ìx1 + 2x2 + 4x3 = 0

ïí5x1 + x2 + 2x3 = 0 ïî4x1 - x2 - 2x3 = 0

ì3x1 - x2 + x3 = 0

ïí2x1 + 3x2 - 4x3 = 0 ïî5x1 + 2x2 - 3x3 = 0

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ì x1 - x2 + 2x3 = 0
7. .а) ï2x + x		2	- 3x = 0
í	1		3
ï	3x + 2x = 0
î	1		3

ì2x1 - x2 - 5x3 = 0 8. .а) ïíx1 + 2x2 - 3x3 = 0 ïî5x1 + x2 + 4x3 = 0

ì5x1 - 5x2 + 4x3 = 0 9. .а) ïí 3x1 + x2 + 3x3 = 0 ïî x1 + 7x2 - x3 = 0

ìx1 + 3x2 - x3 = 0

10. .а) ïí2x1 + 5x2 - 2x3 = 0

ïî x1 + x2 + 5x3 = 0

ì3x1 - x2 + x3 = 0

11. .а) ïíx1 + 2x2 + 4x3 = 0

ïî5x1 + x2 + 2x3 = 0

ìx1 - 2x2 - x3 = 0

12 .а) ïí2x1 + 3x2 + 2x3 = 0 ïî3x1 - 2x2 + 5x3 = 0

ì2x1 + x2 - x3 = 0

13. .а) ïí3x1 - 2x2 + 4x3 = 0

ïî x1 - 5x2 + 3x3 = 0

ì4x1 + x2 + 3x3 = 0

14. .а) ïí 8x1 - x2 + 7x3 = 0

ïî2x1 + 4x2 - 5x3 = 0

б)

б)

б)

б)

б)

б)

б)

б)

16

ìx1 - 2x2 + x3 = 0

ïí3x1 + 3x2 + 5x3 = 0 ïî 4x1 + x2 + 6x3 = 0

ì2x1 + x2 - 3x3 = 0

ïíx1 + 2x2 - 4x3 = 0 ïî x1 - x2 + x3 = 0

ì2x1 - x2 + 2x3 = 0

ïí 4x1 + x2 + 5x3 = 0 ïî2x1 + 2x2 + 3x3 = 0

ì4x1 + x2 + 4x3 = 0 ïí3x1 - 2x2 - x3 = 0 ïî7x1 - x2 + 3x3 = 0

ì 3x1 - 2x2 + x3 = 0 ïí2x1 + 3x2 - 5x3 = 0 ïî 5x1 + x2 - 4x3 = 0

ì2x1 + x2 + 3x3 = 0 ïí3x1 - x2 + 2x3 = 0 ïîx1 + 3x2 + 4x3 = 0

ìx + 2x			- 5x = 0
ï	1	2	3
íx1 - 2x2 - 4x3 = 0
ï		2x - 9x = 0
î		1	3

ìx	- 3x	+ 5x	= 0
ï 1	2	3
íx1 + 2x2 - 3x3 = 0
ï2x - x		+ 2x	= 0
î	1 2	3

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15. .а)

16. .а)

17. .а)

18. .а)

19. .а)

20. .а)

21. .а)

22. .а)

ì x + x		- 3x	= 0
ï 1	2	3
í2x1 + 5x2 + x3 = 0
ïx - 7x		+ 2x		= 0
î 1	2	3

ì x1 - 2x2 + x3 = 0 ïí 3x1 + x2 + 2x3 = 0 ïî2x1 - 3x2 + 5x3 = 0

ì x1 + 2x2 + 3x3 = 0 ïí 2x1 - x2 - x3 = 0 ïî3x1 + 3x2 + 2x3 = 0

ì 3x1 + 2x2 = 0 ïí x1 - x2 + 2x3 = 0

ïî4x1 - 2x2 + 5x3 = 0

ì2x1 - x2 + 3x3 = 0 ïíx1 + 2x2 - 5x3 = 0 ïî 3x1 + x2 + x3 = 0

ì3x1 + 2x2 - x3 = 0

ïí 2x1 - x2 + 3x3 = 0 ïî4x1 + 3x2 + 4x3 = 0

ìx1 - 3x2 - 4x3 = 0

ïí5x1 - 8x2 - 2x3 = 0 ïî 2x1 + x2 - x3 = 0

ì3x1 + 5x2 - x3 = 0

ïí2x1 + 4x2 - 3x3 = 0 ïî x1 - 3x2 + x3 = 0

б)

б)

б)

б)

б)

б)

б)

б)

17

ì 2x1 - x2 + 2x3 = 0 ïí3x1 + 2x2 - 3x3 = 0 ïî 5x1 + x2 - x3 = 0

ì2x1 - x2 + 3x3 = 0 ïíx1 - 3x2 + 2x3 = 0 ïî x1 + 2x2 + x3 = 0

ìx1 - 3x2 - 2x3 = 0

ïí3x1 - x2 + 4x3 = 0 ïî2x1 - 2x2 + x3 = 0

ì5x1 + x2 - 2x3 = 0

ïí 3x1 - x2 + x3 = 0 ïî2x1 + 2x2 - 3x3 = 0

ì3x1 + 2x2 - 3x3 = 0 ïí 2x1 - 3x2 + x3 = 0 ïî 5x1 - x2 - 2x3 = 0

ì4x1 - x2 + 5x3 = 0

ïí2x1 - 3x2 + 2x3 = 0 ïî2x1 + 2x2 + 3x3 = 0

ìx1 + 5x2 + x3 = 0

ïí2x1 - 3x2 - 7x3 = 0 ïî3x1 + 2x2 - 6x3 = 0

ì3x1 + 4x2 - x3 = 0 ïíx1 - 5x2 + 2x3 = 0 ïî 4x1 - x2 + x3 = 0

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23. .а)

24. .а)

25. .а)

26. .а)

27. .а)

28. .а)

29. .а)

30. .а)

ì3x1 - 2x2 + x3 = 0

ïí2x1 - 3x2 + 2x3 = 0 ïî 4x1 + x2 - 4x3 = 0

ì7x1 + x2 - 3x3 = 0

ïí3x1 - 2x2 + 3x3 = 0 ïî x1 - x2 + 2x3 = 0

ìx1 + 2x2 - 4x3 = 0

ïí2x1 - x2 - 3x3 = 0 ïî x1 + 3x2 + x3 = 0

ì7x - 6x					+ x	= 0
ï		1		2	3
í		4x1 + 5x2 = 0
ïx		- 2x	2	+ 3x		= 0
î	1				3

ì5x - 4x + 2x = 0
ï	1	2	3
í	3x2 - x3 = 0
ï	4x	+ x	- 3x = 0
î	1	2	3

ì6x1 + 5x2 - 4x3 = 0 ïí x1 + x2 - x3 = 0 ïî3x1 + 4x2 + 3x3 = 0

ì8x1 + x2 - 3x3 = 0

ïí x1 + 5x2 + x3 = 0 ïî4x1 - 7x2 + 2x3 = 0

ìx1 + 7x2 - 3x3 = 0

ïí 3x1 - 5x2 + x3 = 0 ïî3x1 + 4x2 - 2x3 = 0

б)

б)

б)

б)

б)

б)

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18

ì2x1 + 4x2 - 3x3 = 0 ïí x1 - 3x2 + 2x3 = 0 ïî 3x1 + x2 - x3 = 0

ì 7x1 - 6x2 - x3 = 0 ïí3x1 - 3x2 + 4x3 = 0 ïî4x1 - 3x2 - 5x3 = 0

ì5x1 - 3x2 + 2x3 = 0 ïí2x1 + 4x2 - 3x3 = 0 ïî3x1 - 7x2 + 5x3 = 0

ì x1 - 8x2 + 7x3 = 0 ïí3x1 + 5x2 - 4x3 = 0 ïî4x1 - 3x2 + 3x3 = 0

ì5x1 + 8x2 - 5x3 = 0 ïí 7x1 + 5x2 - x3 = 0 ïî2x1 - 3x2 + 4x3 = 0

ì5x1 + x2 - 6x3 = 0

ïí4x1 + 3x2 - 7x3 = 0 ïî x1 - 2x2 + x3 = 0

ì2x1 - x2 + 4x3 = 0

ïí7x1 - 5x2 + 3x3 = 0 ïî 5x1 - 4x2 - x3 = 0

ì2x1 + 2x2 - x3 = 0

ïí5x1 + 4x2 - 6x3 = 0 ïî3x1 + 2x2 - 5x3 = 0

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19

1.2.6 Виконати дії над матрицями

1. AB + 2NT .

2. BC + 2E

3. A×(B + 3CT ).

4. C × AT + 3N

5. A×(2B - CT ). 6. C × B - 5E .

7. A×(3CT - B)

8. A×CT - NT .

9. A×(2B + CT )

10.

BT × AT + N

11. DT M T - 4E

12. D × M + 4E

13.

F × M T - NT

14. F ×(M T + 4D)

15. M × D + 5E

16.

F ×(M T - 2D).

17. M × F T + 3N

18. F × D + 2NT

19.

DT M T + 3E.

20. F ×(3D - M T )

21. H × P + 3N T

22.

S × P + 3E

23. H × (P - 2ST )

24. S × H T + 2N

25.

H ×(3P - ST )

26. S × P - 3E

27. H ×(2ST - P)

28.

H × ST - NT

29. H ×(3P - ST )

30. PT × H T - N

æ 4 5 -1 2ö

æ3 1 -2 0ö

æ 1 2 4 1 ö

A = ç

÷

, F = ç

÷ , H = ç

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-2 0 3 0ø

è 0 2 3 1 ø

è-3 0

-2 1ø

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2

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S = ç

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1

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1 3ö

æ

0 -1 3 1ö

æ3 -2 5 1 ö

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M =

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C = ç

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0 1 3 -1÷

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-1 4

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1 5 4 0

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-1 0 1

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4 -1 0

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20

1.2.7. Знайти значення многочлена f ( A) від матриці А , коли

f (x) = x2 - 4x + 3 і матриця А дорівнює:

æ 1 1 -2

ö

æ

2 0 1ö

3.

æ

0 4 5 ö

æ 1 1 -2ö

1. ç

0 1 -1

÷

2. ç

1 -2 3

÷

ç

1 1 -3

÷

4. ç

-2 0 3

÷

ç

÷

ç

÷

ç

÷

ç

÷

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