Матрицы и определители

0	0 1	1		0	0	1	1 0		0
B A =			=					≠	.
1	1 0	0		1	1	0	0	1	1
! * ( *			*, " A B ≠ B A (
A B . 5 .		2 *		, * )$! % !

& % * * ! ! 4 *:

An En = En An = An .

)* ,. : 1. (A B ) C = A (B C ).

2. A (B + C ) = A B + A C . 3. (A + B ) C = A C + B C .

4. α (A B ) = (αA) B = A(αB ).

! *, " A, B, C		,	.
	%
! & –	m × n .	,
	n × m * ,. *: / aij " (i, j )
* . " ( j, i ), . . i -		j -	j -,

i - . 3 , * , , % &,

,	–		% !			& " , T .
2 *	,
			7	2	0		7		− 1
						=
	A =				, AT			2	3		.
		− 1
				3	− 5			0
									− 5

* * / *

, –	.
)* : (A B )T = BT AT .
1.3. , % 1 # *
	− 1 2 7			=	1 − 1 − 3
+ * 1.1. ! A =			, B				.
	3	0				2
			4		0		5
2 2 A + 3B .
	− 1 2 7				1 − 1 − 3
, . 2 A + 3B = 2			+ 3				=
	3	0			0	2
			4				5

11

	2(− 1)		2 2 2 7 3 1 3 (− 1) 3 (− 3)
	=									+						=
		2 3	2 0 2										2		3 5
							4 3 0 3
		− 2 + 3 4 − 3 14 − 9 1 1 5
		=										=			.
			6 +
					0 0 + 6 8 + 15 6 6 23
				2 − 1							− 7		3
+ * 1.2. ! A =							; B =						.
						− 4						1
				8									0
2	AT − 2B .
, .
		AT					2 − 1 T						− 7 3
					− 2B =						−		2		=
														1
							8				− 4				0
			2			8					− 14 6 16				2
		=						−					=		.
														− 3
			− 1 − 4								2 0				− 4
					2	− 3					− 3		8
+ * 1.3. ! A =
					0	1		, B =					.
					5	4					− 1		0
2	A (− B ).
, . '										*							*
A (− B )	.,					[3 × 2][2 × 2]= [3 × 2],											A (− B )
[3 × 2]:
								2			− 3		− 8
			A		(− B ) =				0		1	3
														=
									5		4	1	0
	2 3 + (− 3) 1 2 (− 8)+ (− 3) 0														3 − 16
						0 (− 8)+ 1 0
	= 0 3 + 1 1													=	1	0 .
		5 3 + 4 1				5 (− 8)+ 4 0
													19 −			40
+ * 1.4. 2 A B C ,
				2										3	2
		A = − 1 , B =					(1 2 3), C = − 1								1 .
				0										0
															− 1
, .		!									A B C			.,
[3 ×1][1× 3][3 × 2]= [3 × 2]															3 × 2 . % "
											12

*	A B C					*
(A B ) C = A (B C ) "				B C , A (B C ):
				3		2
		B C = (1 2 3) − 1				1 =
				0		− 1
	= (1 3 + 2 (− 1)+ 3 0 1 2 + 2 1 + 3 (− 1)) = (1 1),
			2			2	2
	A (B C ) =		− 1 (1 1) =			− 1 − 1 .
			0			0
							0

+ * 1.5. 2 (A B )T ,
	2 − 3			− 3 1 − 2
A =		;	B =		.
	1 4			2 0
					1

, .	)	"			(A B )T = BT AT , "
- *			:
) (A B )T		2 − 3		− 3 1 − 2 T		=
	=
		1
			4	2 0	1

2 (− 3)+ (− 3) 2 2 1 + (− 3) 0 2 (− 2)+ (− 3) 1 T
=					=
	(− 3)+ 4	2	1 1 + 4 0	1 (− 2)+ 4 1
1

				− 12						− 7 T				− 12			5
							2
				=							=					2	1	;
						5	1			2						− 7	2
		− 3		2							1
) BT	=		1	0		; AT =				2
													,
										− 3	4
			− 2
				1
									− 3		2						1
				BT AT			=			1 0						2
																	=
															− 3
									− 2								4
											1

	(− 3) 2 + 2 (− 3) (− 3) 1 + 2 4				− 12	5
	1 2 + 0 (− 3)
=		1 1 + 0 4		=	2	1 .
	(− 2) 2 + 1 (− 3) (− 2) 1 + 1 4				− 7
						2
		13

					1.4. + * ( ! $
* (.
	− 1	2		0	3
1.		−			.
	7			− 5
		3			4

	1	− 2	3			0	1 T
				− 3			− 1
2.	2					2			.
	1	0	4			1
							1

14

		− 1
		2
	3. (2 0 − 1 3)	3	.
		1

15

		2
4.		0		(− 1 2 3 1).
		− 1
		3

5. 2 A B B AT ,

7		0	1				1	0	7
					, B =		− 1
A =								3	0		.
	3	2	− 2
							2	5	2

16

17

6. 2 A BT ,	− 1
	A =
		0

		− 7
2
	, B =		5
3			1
			0

3

8

4 .

− 2

18

2.

)$! % ! & * ", " * * *

. ' * *

, % ' , ( '

. . " , * A , det A

	a11		a12		...	a1n
	a21		a22		...	a2n			.	(2.1)
		.		.	.		.
	an1		an 2		...	ann
! * *								n - *,		" ,.
a			a		...	a
	11		12				1n
A = a		21	a	22	...	a	2n	.
					...
an1			an 2			ann

2.1. % ' ( ' %

	1.				*		*, " ,.
a	b		", * , /
1	1	,
	b2
a2
					* /						* " ,
. .
						a1	b1		= a b		− b a	2	.	(2.2)
						a2	b2		1	2	1
	2 *		,	2 − 7 = 2 5 − (− 7) 3 = 31.

35

2. * *, " ,.

a		b
	1	1
A = a		b
	2	2
a3		b3

",

c1

c2 , c3

19

a1	b1	c1
a2	b2	c2	= a1b2 c3 + b1c2 a3 + c1a2b3 − c1b2a3 − b1a2c3 − a1c2b3 .	(2.3)
a3	b3	c3

0 , " (2.3) *

" *, * – " ". 0 * (2.3) *

. ' «* ,»

* / , . ( -(

, * ( . 2.1). # « » * , ",

* " ( . . 2.1).

			. 2.1
* 3- *			"				*		− *
* ) ,.
% / ). ' * ) ,								&
		, , * *				&	* "
*, *		. 0		*		/
			(	,	*			.		2.2
(* , / * ),
* "-( - * " ,						. ! /
* /		, (				«-», ,

* *.

. 2.2

2 *, " * " * ( '

* * , * * ) , * " * " (2.3).

20