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Solution. We take a minor of the second order, for example,

Μ(2) =	1 −2	= 1 ≠ 0.

	−1 3
The bordering minors of the third order for its are

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

If both minors of the third order are equal to zero and the minor M (2) of the second order isn’t equal to zero, then by the definition r(A) = 2.

2		−2	1	−1	3
	−4	−4	2	−4	−2
Example 6. Find the rank of matrix A =	−4	−4	2	−4	−2	.
	3	0	0	2	3

	−2	−6	3	−5	1
	−2	−6	3	−5	1

Solution. Let’s consider the elementary transformations to the given matrix

2		−2	1 −1		3	2			0	1 −1 3		1			0	3	−1 3
	−4	−4	2	−4 −2				−4	0	2	−4 −2			2	0	−2	−4 −2
							~						~					~
	3	0 0		2 3				3	0	0	2 3			0	0	3	2 3

	−2	−6	3	−5	1			−2	0	3	−5 1			3	0	1	−5 1

1			−1 3	0		0	1			−1 3 0		0
	2		−4 −2 0			0		0		−2	−8 0	0
~							~								~
	0		2 3	0		0		0		2 3 0		0

	3		−5 1	0		0		0		−2	−8 0	0

		1	−1 3		0	0	1 −1 3 0						0
		0	−2 −8		0	0			0	−2	−8 0		0
~							~							.
		0	2 3		0	0			0 0		−5 0		0

		0	0 0		0	0			0 0 0 0				0

The determinant of the third order, which consists of the cross elements of the first 3 rows and columns of the last matrix, is not equal to zero, and all the

minors of the forth order are equal to zero. So, r(A) = 3 .

Micromodule 2

CLASS AND HOME ASSIGNMENT

Calculate the product of the matrices:

					−2			3	1		4		2
1.	AB and	BA , if			−2			3	1	,		7	3
1.	AB and	BA , if		A =		5		0	6	,	B =	7	3	.
						5		0	6			−1	1
												−1	1
		1		2	−1					6
2.			3 0 1				,	B =
2.	AB , if A =		3 0 1				,	B =		−3 .
			−2	1		4				9
			−2	1		4				9
			4 3					−4 −2 4						2		3	−1
3.	ABC , if		4 3			B =		−4 −2 4				, C =			3	1	0
3.	ABC , if	A =			,	B =						, C =			3	1	0	.
			2 1						1 0 3						−2	0	0
															−2	0	0

Find f (A) if:

4.A =

5.A = 1−2

−3	, f	(x) = (x2 − 3x) (3x + 2) .
5
0	4
3	−2 , f (x) = 2x2 + x − 4 .
1	2
1	2

Find an inverse matrix, if:

		1	1	−1			1	3	−1
6.		0	−2 −3		. 7.		1	1	−5
	A =					A =				.
		2	3				−1 −2 2
				−1

Find the rank of matrix:

	1		−2	3		1 −2		1	−1 3
		3	2	4		−2	−4	2	−4	0
8.		3	2	4	. 9.	−2	−4	2	−4	0	.
		−2	−4	−1		−5	−4	5	−2	3
		4 0		7		−6	−10 8		−7	6
		4 0		7		−6	−10 8		−7	6

Answers

					12	6				2	12		16		−9			−242	−66
										2	12		16

	1.		AB =				,	BA =		1 21 25 . 2. AB =						27 . 4.	f (A) =	132	−176	.
				14		16				7	−3							132	−176
										7	−3		5			21
			11		−2		−5
6.	A	−1		−6		1	3
6.	A		=	−6		1	3	.
				4	−1		−2
				4	−1		−2
											Micromodule 2
	2.1. Find f (A) if:								SELF-TEST ASSIGNMENT
	2.1. Find f (A) if:
					1		2	−1
	2.1.1.					3	0	1		f (x) = 2x				2	+ 3x + 1 .
	2.1.1.			A =		3	0	1	,	f (x) = 2x					+ 3x + 1 .
						0	1	0
						0	1	0
					3		4		0
	2.1.2 . A = 1						2		3 ,		f (x) = x2 + 5x + 2 .
					−3		1 −1
					2		1	,	f (x) = x			3	+ x	2	− 7x + 3 .
	2.1.3. A =					0		,	f (x) = x				+ x		− 7x + 3 .
						0	−1
					1		−2	3
	2.1.4. A =					2	−4	1 ,		f (x) = 3x2 − 2x + 8 .
						3	−5 2
						−2	2	−1
	2.1.5. A =					1	0		5 ,		f (x) = x2 − 3x − 1 .
					4		3		0
					2		1	1
	2.1.6. A =					3	1	2 ,		f (x) = x2 − x − 1 .
					1		−1 0
					−1		3		0
	2.1.7. A =					−2	1		1 ,		f (x) = x2 − 2x + 1.
					3		0	−2
																			33

2.1.8. A =

2.1.9. A =

2.1.10. A =

2.1.11. A =

2.1.12. A =

2.1.13. A =

2.1.14. A =

2.1.15. A =

2.1.16. A =

2.1.17. A =

2.1.18. A =

−3 4

−3 2

−10

−4

−2

−2−1

−10

1		3
0 −1 , f (x) = −2x2 + 8x − 6 .
2
2	−1
1		f (x) = x	3	− 4x .
,		f (x) = x		− 4x .
1
1		4
3 0 , f (x) = 2x2 + 6x − 3 .
−2
−2		3
8		1
2 3 , f (x) = x2 − 5x + 3 .

−3 −1
−3 −1
2		3 , f (x) = 5x2 + 2x − 8.
1
1		2
2	, f (x) = x3 − 4x2 + x − 1.
1
−1 , f (x) = 2x3 − 8x + 6 .
2
3		1
1 0 , f (x) = x2 − 6x + 9.
5
5		−4
3		0
−1 5 , f (x) = 9x2 − 4 .
2
2		−1
−2		1
3		6 , f (x) = 9 − x2 .
4
4		−5
0		4
3 0 , f (x) = 2x2 + 8x + 8 .
−1
−1		2

	1		3	2
2.1.19.		−4	0	5			,	f (x) =			5 − 4x − x						2	.
2.1.19.	A =	−4	0	5			,	f (x) =			5 − 4x − x							.
		2	4	−7
		2	4	−7
	5		4	0
2.1.20.		1	3	1			,	f (x) =			3x		2	− 7x + 5 .
2.1.20.	A =	1	3	1			,	f (x) =			3x			− 7x + 5 .
		−2	1	−4
		−2	1	−4
	6		2	5
2.1.21.		−1	3	1		,		f (x) = (x + 3)							2	.
2.1.21.	A =	−1	3	1		,		f (x) = (x + 3)								.
		0	4	2
		0	4	2
	3		2	1
2.1.22.		4	1	2		, f (x) = (x − 2) (2x + 3).
2.1.22.	A =	4	1	2		, f (x) = (x − 2) (2x + 3).
		1	−5	3
		1	−5	3
	8		0	2
2.1.23.		−3	2	−3			,	f (x) =			4 −			2x − x			2	.
2.1.23.	A =	−3	2	−3			,	f (x) =			4 −			2x − x				.
		1	1
		1	1	−1
	3		−1	4
2.1.24.		0	2	5		,		f (x) =		5x		2	− 8x + 6 .
2.1.24.	A =	0	2	5		,		f (x) =		5x			− 8x + 6 .
		5	−1	8
		5	−1	8
2.1.25.	1		−5	,	f (x) = 3x				3	+ 4x				2	− 2 .
2.1.25.	A =	6	2	,	f (x) = 3x					+ 4x					− 2 .
		6	2
	2		−3	,	f (x) = (x				2	− 2x)(2x − 1) .
2.1.26. A =		4	5	,	f (x) = (x					− 2x)(2x − 1) .
		4	5
	0		3	5
2.1.27.		−1	4	−2			,	f (x) = (4 − 2x)(x + 6) .
2.1.27.	A =	−1	4	−2			,	f (x) = (4 − 2x)(x + 6) .
		2	1	3
		2	1	3
	7		3	2
2.1.28. A =		−1	4	5 ,				f (x) = 2x2 − 6x + 7 .
	2		1	1
2.1.29.	2		−3	,	f (x) = (x				2	− 2x)(2x + 1) .
2.1.29.	A =	4	5	,	f (x) = (x					− 2x)(2x + 1) .
		4	5

	1		2	4
2.1.30.		1	3	−2	,	f (x) = 5x	2	+ 3x − 4 .
2.1.30.	A =	1	3	−2	,	f (x) = 5x		+ 3x − 4 .
		−1	0	5
		−1	0	5

2.2. Solve the matrix equations.

2.2.1.	A X = B ,	3		2	,	1
		A =	1	1		B =
						0
		4		3		2
2.2.2.	X A = B ,				,		1
		A =	1	1		B =
							0

2 7

4 8 .

1 . −1

						2		3							5			4	1				1	1	0
2.2.3.	A X B = C , A =					2		3		,	B =				1			1	7		, C =		1	1	0
2.2.3.	A X B = C , A =									,	B =				1			1	7		, C =			1	.
						5		8							6			5	9				0	1	1
															6			5	9
2.2.4.	A X = B ,	A =		2			1		,	B =			1			0		7
2.2.4.	A X = B ,	A =			1				,	B =				8		1			.
					1		1							8		1		2
				3			2						4		7
2.2.5.	X A = B ,	A =		3			2		,	B =				0	1
2.2.5.	X A = B ,	A =			1		1		,	B =				0	1		.
					1		1							2	3
														2	3
2.2.6.						4 3								6 8							=	5 4
2.2.6.	A X B = C , A =								,			B =							, С		=			.
						1 1								2 1								−2 0
				1			3		2						1				0
2.2.7.	A X = B ,	A =			3		8		5		,	B =				3			1
2.2.7.	A X = B ,	A =			3		8		5		,	B =				3			1	.
					2		7		6							3		−4
					2		7		6							3		−4
			4			3		6						2			1		5
2.2.8.	X A = B ,			5		4		7			B =			2			1		5
2.2.8.	X A = B ,	A =		5		4		7	,		B =									.
				6		5		7						0			0		−1
				6		5		7
						5 4									7		8		7			−1		−1 0
2.2.9.	A X B = C ,		A =			5 4				,					8		9		6			−1		−1 0
2.2.9.	A X B = C ,		A =							,	B =				8		9		6	,	C =					.
						9 7									5		6		8				1 1 0
															5		6		8
					1			1			2						1			5
2.2.10. A X = B ,			A =			2		−1			2		,	B =				4
2.2.10. A X = B ,			A =			2		−1			2		,	B =				4		−1 .
						4		1			2							2		7
						4		1			2							2		7

			5		5	3							6			4		6
2.2.11.	X A = B ,			3	4	2		,		B			6			4		6
2.2.11.	X A = B ,	A =		3	4	2		,		B		=		0		3		−5		.
				4	3	5								0		3		−5
				4	3	5		2																	1		2
					3	3		2							2				3						1		2
2.2.12. A X B = C ,			A =		4	5		3			,		B =		2				3			,				4	6
2.2.12. A X B = C ,			A =		4	5		3			,		B =									,	C =			4	6	.
					5	4		4								−3		−4								4	7
					5	4		4																		4	7
			1 2 −3											1 1					1
2.2.13.	A X = B ,			0	1	2					B =				2		−3		1
2.2.13.	A X = B ,	A =		0	1	2		,			B =				2		−3		1		.
				1	0	4									4		1		5
				1	0	4									4		1		5
			2 1 3										1 −1 3
2.2.14.	X A = B ,			0	2	0		,		B				4			2		0
2.2.14.	X A = B ,	A =		0	2	0		,		B		=		4			2		0	.
				1	4	5								1			4		5
				1	4	5								1			4		5
2.2.15. A X B = C ,			A =		3 4		,		B =				−1 2						,						1 0
2.2.15. A X B = C ,			A =				,		B =					3			−5		,	С =						.
					1 1									3			−5								0 1
2.2.16. X A B = C ,			A =		3 −2				, B				3 2						С	=		2 1
2.2.16. X A B = C ,			A =						, B			=					,		С	=						.
					1 −1									1 4								4			−3
			5		3			4
2.2.17.	X A = B ,			−6	−3			−5				,	B = (3					2		1) .
2.2.17.	X A = B ,	A =		−6	−3			−5				,	B = (3					2		1) .
				4	2			2
				4	2			2
2.2.18. A X B = C ,			A =		4 5					,				5 8							,		С		2 4
2.2.18. A X B = C ,			A =		−5 −6					,		B =						−3			,		С		=		−4		.
					−5 −6									−2				−3							−3		−4
				2	4	5								1				5			1
2.2.19.	X A = B ,			3	−1	1			,		B =				−3		−1
2.2.19.	X A = B ,	A =		3	−1	1			,		B =				−3		−1				−1 .
				1	1	1									2			5			3
				1	1	1									2			5			3
2.2.20. A X B = C ,			A =		3 −1				,				5 6						С =				14 16
2.2.20. A X B = C ,			A =		5 −2				,		B =						,		С =					9 10			.
					5 −2								7 8											9 10
2.2.21. A X B = C ,			A =		2 1		,						−3 2						,						−2 4
2.2.21. A X B = C ,			A =				,		B =					3					,	С =							.
					3 2									3			−1								3 −1
			5 3 1													−8 3 0
2.2.22.	X A = B ,			1	−3			−2				,	B =				−5		9				0
2.2.22.	X A = B ,	A =		1	−3			−2				,	B =				−5		9				0	.
				−5 2 1													−2 15 0
				−5 2 1													−2 15 0

		1 2			−3		1 0		−3
2.2.23.	A X = B ,		3	2	−4	,		2	7
		A =					B = 10			.
			2	−1 0				7	8
							10

	1		1	1		1	1	2
2.2.24. A X = B ,			2	3	,		1	2
	A = 1					B = 1			.
			3	4			1	2
	1					1
	2		3	1
2.2.25. X A = B ,		3	7	2		, B = (10		3	3) .
	A =
		5	4	2

2.2.26. A X B = C ,				1			2			,			1		3		2
2.2.26. A X B = C ,			A =		−3					,	B =					,	С =	−6
					−3		−5				−1 −2							−6
			2			2	3							1	0
2.2.27.	A X = B ,			1		−1	0			,	B =			3	1
2.2.27.	A X = B ,	A =		1		−1	0			,	B =			3	1	.
				−1 2 1
				−1 2 1									−3 −4
			1 1 −1										1 −1			3
2.2.28.	X A = B ,			2		1	0			,	B =		4		3	2
2.2.28.	X A = B ,	A =		2		1	0			,	B =		4		3	2	.
				1	−1 1								1 −2 5
				1	−1 1								1 −2 5
2.2.29. X A B = C ,				2		2					−1			4				−2)
2.2.29. X A B = C ,			A =			1	,	B =						−7	, С = (1			−2)
				3		1					2			−7
			5		−6		4				3
2.2.30.	A X = B ,			3		−3	2		,			2
2.2.30.	A X = B ,	A =		3		−3	2		,		B =	2		.
				4		−5	2					1
				4		−5	2					1

4 −11 .

2.3. Find the inverse matrix, if:

	1		3	−1
2.3.1.		0	−6	−5
	A =				.
		1	2	−2

	−1		2	−2
2.3.3.		−3	5	−4
	A =				.
		1	0	1

	1		−1	−2
2.3.5.		0	−5	−4
	A =				.
		2	−7	−9

	1		5	−1
2.3.2.		−1	4
	A =			−3 .
		−1	0
				−1
		2	−2	1
2.3.4.		0	−5	−3
	A =				.
		2	−1	2

		2	1	−3
2.3.6.		0	−3	−2
	A =				.
		2	−2	−4


	1		1	−1
2.3.7.		0	−7	−3
	A =					.
		3	3	−4

	2		2	1
2.3.9.		−3	−3	3
	A =				.
		−1	0	1

	1		1	−3
2.3.11.		4	7
	A =			−11 .
		1	−1	0

	1		2	−1
2.3.13.		6	1
	A =			−1 .
		8	1	0

	1		−1	3
2.3.15.		3	−6	3
	A =				.
		1	5	0

	1		2	11
2.3.17.		4	−3	−2
	A =					.
		1	−1	0

	1		−2	1
2.3.19.		2	−3	0
	A =				.
		2	6	1

	−1		7	0
2.3.21.		1	9	3
	A =				.
		3	2	3

	−1		0		3
2.3.23.		3	−1		9
	A =					.
		4
			−5 11
	9		−2	1
2.3.25.		2	−5	0
	A =				.
		4	−2	1

3			2		−1
	−1		−1
2.3.8. A =					−1 .
	−2		0	−2

	1		2		1
2.3.10. A =		3	5		1
						.
		2	−2		0

	1		1	4
2.3.12. A =		4	7	3
					.
		2	2	0

	2		1		1
2.3.14. A =		5	8		1
							.
		−1	−7		0

	1		−1		4
2.3.16. A =		2	−3		4
						.
		4	−4		0

	3		2		3
2.3.18. A =		−5	6		1
						.
		−3	3		0

	1		1		−3
2.3.20. A =		4	−8		4
							.
		5	−5		0

		4	11		0
2.3.22. A =		1	13
					−1 .
		1	−2		1

	5		1	2
2.3.24. A =		1	4	0
					.
		2	1	2

	1		0		1
2.3.26. A =		4	−13		4
							.
		5	−3		3

				1			0	1
2.3.27.						−3	9
2.3.27.		A =				−3	9	3 .
						0	2
						0	2	3
				2			−10	7
2.3.29.		A =			0		29	7
2.3.29.		A =			0		29	7	.
					1		−5	4
					1		−5	4
2.4. Find the rank of matrix.
	0					1	−1		2			2
		−2				1	3	−1				1
2.4.1.		−2				1	3	−1				1			.
		3			−2		1		1			−2
		1				0	3		2			1
		1				0	3		2			1
	−2				2		1	−1	−3
		3			1		0	1		0
2.4.3.		3			1		0	1		0		.
		−1			1		2	2		1
		0			4		3	2
		0			4		3	2		−2
	3				1		3	2		5
		−1			2		3	1		0
2.4.5.		−1			2		3	1		0
		0			1		−1	2		2
		2			4		5	5		7
		2			4		5	5		7
	2					1	−2		1			3
		1			−1		0		2			0
2.4.7.		1			−1		0		2			0	.
		−1				2	1	−1				1
		2				2	−1		2			4
		2				2	−1		2			4
	−1				0		1	−1			0
		−2			1		−2	1			−4
2.4.9.		−2			1		−2	1			−4			.
		2			3		−1	1			1
		−1			4		−2	1			−3
		−1			4		−2	1			−3
		4				1		2	−3					−5

2.4.11.			1			−2		1		0				−1 .
			−3			−1 −2				1				3
			2			−2		1	−2					−3
			2			−2		1	−2					−3

	2		−2	13
2.3.28.		0	−1		4
	A =					.
		2	−2	1

	1		2	1
2.3.30.		0	5	1
	A =				.
		1	2	3

	1			1 −2		1	3

2.4.2.		−1		2	1	−1 −2 .
		3		0	2	1	−1
		3		3	1	1
							−1
	−1			−1	1	−1	0
					−1
2.4.4.		2		1		3	1 .
		3		−1	−1	2	−1
		4		−1	−1	4
							0
	1			1	1	−1	2
		−1		1	−1	−1	0
2.4.6.								.
		1		−1	2	3	0
		1		1	2	1	2

	3			−1	2	3	5
		−2		1	−1	−2	−3
2.4.8.									.
		1		2	1	−1	2
		2		2	2	0	4

		1		−2	2	−1	4
			3	1	−1	1	−2
2.4.10.									.
			2	0	3	1	3
			6	−1	4	1	5

		1		−1	2	−1		−3
			3	2	−1 1			2
2.4.12.										.
			−2	1	1	2		1
			2	2	2	2		0

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