Матрицы и определители

ab

9. .

− b a

cosα − sinα

10. . sinα cosα

	− 2	0	2
11.	2	− 1	− 2	.
	1	3	− 1

31

6 4 2

12. 7 5 1 .

3 2 4

32

	2	1	1
13.	1	− 4	− 1	.
	− 1	8	3

	8	6	10
14.	5	7	3	.
	4	3	3

33

		2	− 3	4	− 1
	15.	5	2	− 1	0	.
		4	− 1	2	1
		0	− 2	1	− 1

34

35

		5	− 3	4	2
	16.	− 2	2	3	4	.
		3	− 3	5	3
		2	4	− 3	3

36

37

3.2

3.1.. /03 % % $ !

! & − * n, E – "

* n:

a	a	...	a		1	0 ...	0
11	12		1n
a21	a22	...	a2n		0	1 ...	0
A = .	.	.	.	,	E = . . .		.	.	(3.1)
	an 2	...				0 ...	1
an1			ann		0

1. ',

,.

B A = A B = E ,

A " A−1 .

*

*

(* ) , "

* n.

2

, A−1 .

%

.

$1 *

,

" *

&

" .

2.

% .

!,

* " , % .

!,

A

= 0 .

3

,

A−1

.

A .

10 * (

A B

=

A

B

)

, "

A−1

=

1

≠ 0 .

A

A A−1 = E

A A−1

A

A−1

%,

=

E

= 1;

A−1

=

1

.

A

:

(A−1 )−1 = A,

(A B )−1 = B −1 A−1 ,

(AT )−1 = (A−1 )T .

3.2. * $ !

& ( * *

3- *.

38

1.

2 (

*

A

. 4

A

= 0 ,

.. ' " -,. 4

A

≠ 0 ,

*

(

.

2.2 ( " * Aij ( /

aij

&.

3.

)

" (

*

* ,

~

A :

A

A

A

~

11

12

13

A

=

A

A

A

.

21

22

23

A31

A32

A33

4. 3 *

~

A :

A

A

A

~T

11

21

31

A

= A

A

A

.

12

22

32

A13

A23

A33

5.

,

A−1 :

A

A

A

11

21

31

A

A

A

A−1

1

~

A

A

A

=

AT =

12

22

32

.

(3.2)

A

A

A

A

A

A

A

13

23

33

A

A

A

1

− 2

3.1. 2 A−1 , A =

.

5

− 3

, .

1. A = 1 − 2 = 7 ≠ 0 .

5− 3

2.A11 = −3 ; A12 = −5 ;

	A21 = 2 ;			A22 = 1 .
3.	~	− 3			− 5
	A =					.
			2		1
4.	~T		−	3	2
	A	=				.
				5	1
			−

39

5. A−1											1		− 3 2 − 3 7 2 7
							=															=									.
																								− 5 7 1 7
											7 − 5 1
			%.
			A A−1												1 − 2 − 3 7 2 7 1 0
														=																	=						= E .
																5 − 3 − 5 7 1 7 0 1
			%: A−1 =																		1	− 3			2
																									.
																					7 − 5				1
																																			1	− 3					1
			3.2. 2 A−1 , A =																																0		1				3 .
																																			0		− 1
																																									3
			, .
														− 3			1							1	3
						=		1												= 1								= 6 ≠ 0 .
1.			A					0 1 3
								0						− 1			3							− 1	3
															3														0 3									0 1
2. A =										1							= 6 ;							A = −									= 0 ; A =										= 0 ;
	11									− 1					3									12					0		3					13		0			− 1
																										1 1														1 − 3
	A = −											− 3 1						= 8 ; A =													= 3;				A = −									= 1 ;
	21											− 1					3							22		0			3							23				0 − 1
	A =								− 3 1								= −10 ; A = −													1 1			= −3; A =						1 − 3				= 1.
	31									1					3									32						0	3					33			0		1
										6							0					0
	~
3.			A			=				8							3					1 .
																− 3
							− 10															1
										6					8			− 10
4.			~T												3
			A				= 0												− 3 .
															1				1
										0
																	1					4 3			− 5 3
				−1						1				~T
5.			A				=						A			= 0 1 2 − 1 2 .
										6
																			0			1 6			1 6

40