# Введение в теорию определителей: Методические указания по курсу ''Алгебра и геометрия''

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08-Dec-2016

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• 506

1 / /

: . .

. ..

2001

• 2

.

-, . - " " , - . , . " ", - , .

, . , - . . . 20 . - . .

1 2- 3- .

. -

=+=+

.,

222

111cybxa

cybxa (1)

(1)

.22

11

=

baba

A (2)

2- . (1) - :

.2

1

=

cc

c (3)

, (2) (3) (1).

x (1) 2b , 1b .

1

2

222

111

,,

bb

cybxacybxa

=+=+

( ) .12211221 bcbcxybba = y

( ) .12211221 cacaybaba = , 01221 baba . - x y :

• 3

1221

1221bababcbc

x

= , .1221

1221babacaca

y

= (4)

1221 baba , (4) - :

1) 21ba , 2)

12ba . - +

1221 baba 2-

Det(A)= .|| 212122

11 abbababa

A ==

(4) . , c1b2-b1c2 x A1, - :

Det(A1)= .|| 212122

111 cbbcbc

bcA ==

y A2, -

Det(A2)= .|| 212122

112 accaca

caA ==

C (4)

11xT y (1)

,

22

11

22

11

bababcbc

x = .

22

11

22

11

babacaca

y =

3-

=++=++

=++

.,

,

3333

2222

1111

dzcybxa (5)

=

333

222

111

cbacbacba

A , .3

2

1

=

ddd

d

• 4 x 1- 3232 bccb , 2- 3113 cbcb , 3- 1221 cbcb .

=++ xcbacbacbacbacbacba )( 123312231213132321 .123312231213132321 cbdcbdcbdcbdcbdcbd ++=

, x 0. -

.123312231213132321

123312231213132321cbacbacbacbacbacbacbdcbdcbdcbdcbdcbd

x++++

= (6)

, (6), :

+ 3- , , ;

- 2- , - , 2-

a1 b1 c1 a1 b1 c1 a2 b2 c2 , a2 b2 c2 . a3 b3 c3 a3 b3 c3 + _

det(A)=|A|= cbacbacbacbacbacba 23312231213132321 ++ 3- . (6) - .

=

333

222

111

1

cbdcbdcbd

A ,

. , (6) x

:

|||| 1

AA

x = .

y z:

|||| 2

AA

y = , |||| 3

AA

z = ,

• 5

=

333

222

111

2

cdacdacda

A ,

=

333

222

111

3

A

A 2 3- - d.

1. 3-

3006081102210321

=++= .

2 n- . .

, n- . n-

=

nnnn

n

n

aaa

aaaaaa

A

21

22221

11211

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

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

n2 . , .

2 - 3 . :

1) ;

2) - ;

3) +, -. n ,

2 ? - . n . . , - , . - , . - (n-1) . - 1. - . n(n-1) 1=n! c . , - .

n n! - ,

• 6 . + - .

2 4-

44434241

34333231

24232221

14131211

aaaaaaaaaaaaaaaa

.

1) a11a23a34a43 a12a23a34a41a32 ? - . 2) a12a31a22a44 ? - . 3) a11a22a33a44 a21a43a14a32? , .

? -

,

211222112221

1211 aaaaaaaa

= .

. + a11a 22 - (1,2), - a12 a21 - (2,1).

322311332112312213322113312312332211

333231

232221

131211

aaaaaaaaaaaaaaaaaaaaaaaaaaa

++= .

, - . :

+ - (1,2,3) (3,2,1),

(2,3,1) (2,1,3), (7) (3,1,2) (1,3,2).

, + , , - , - .

1 1 3 2 2 3

• 7 3 .

- , . ,...,, 1,2, ,n.

1. , , , - , .

2. , , -

. 3. ,

, - .

(7), + , - . , - +, , .

.

nnnn

n

n

aaa

aaaaaa

A

21

22221

11211

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

...........................|| = .

naaaP ....21= .

, . , - . - , ( ) ,...,, , - (1,2, ,n). +, ( ) ,...,, - -, - . ,

4. - A n n!

naaatP ....21)1(= ,

( ) ,...,, - (1,2, ,n), t - .

= =),..,(

,...,21

naaap

A P .

• 8 , -

.

4. . ( ) ,...,, (1,2,..,n). 2 , - . (1,2,3,4,..,n) (2,1,3,4,..,n).

5. , , .

1.

- .

3. (4,1,3,2). - (3,1,2,4).

1 3 (4,3,1,2); 4 3 (3,4,1,2); 4 1 (3,1,4,2); 4 2 (3,1,2,4).

1. . n=2 - : (1,2) (2,1). , k=2, n-1 k=n. 2 - n :

( ) ,...,, ( ) ,...,, . 2 :

) = . ( ) ,...,, ( ) ,...,, n-1 . - . ) .

) . : ' , - . .

1. - , . . - . . . ( ) ,...,,,,,...,, ( ) ,...,,,,,...,, . ,...,, A, - ,...,, - B.

( ) ,...,,,,,,...,, ; (8)

• 9 ( ) ,...,,,,,,...,, . (9)

. ( ) ( ) ( )Aaaa ,,,, . - . ( ) ( ) ( )Bbbb ,,,, . ( ), ( ), . , (8) (9) - . , (8) (9) . 1 - .

. ( ) ,...,,,...,,,..., . k . 1 k . - ( ) ,...,,,,...,,..., . k - . ( ) ,...,,,...,,,..., . - ( ) ,...,,,,...,,..., . . 1 k ( ) ,...,,,...,,,..., . 2k+1 . . .

. 1.1. -

, .

1.2. ( )n ,...,, 21

, ( )n,...,2,1 .

2. -

2!n .

. a - , b - a+b=n!. 2 . 1- ( ), 2- - (b ). 1- . . ( 1.), . b. - ba . , , - . - ab . ba ab , ba = ; 2*a=n!; a=n!/2 .

5. .

• 10

=

nnnn

n

n

aaa

aaaaaa

A

21

22221

11211

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

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

A.

=

nnnn

n

n

aaa

aaaaaa

A

21

22212

12111

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

...........................' .

A .

2. . .

=

nnnn

n

n

aaa

aaaaaa

A

21

22221

11211

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

........................... -

. |A| |A|. -

nnaaaP 2211= , ( )n ,...,, 21 - -

1,2, ,n. |A| |A|. , |A| |A| - .

, - . ( )n ,...,, 21 . - ( )n ,...,, 21 - . - bij=aji,

=

nnnn

n

n

bbb

bbbbbb

A

21

22221

11211

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

...........................' .

P nn nn bbbaaaP 2121 2121 == .

nnbbb 21 21 . , - :

nnbbbP 21 21= . -

• 11 , ( )n ,...,, 21 ( )n,...,2,1 - . ( )n,...,2,1 ( )n ,...,, 21 . 1.2 , ( )n ,...,, 21 - . , - . ( )n,...,2,1 ( )n ,...,, 21 . . , - ( )n ,...,, 21 . - ( )n ,...,, 21 . P |A| , |A|. .

3. , , .

. : i j ( ji ).

nnnn

jnjj

inii

n

aaa

aaa

aaa

aaa

A

21

21

21

11211

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

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

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

|| = ,

nnnn

inii

jnjj

n

aaa

aaa

aaa

aaa

B

21

21

21

11211

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

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

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

|| = .

nn

ajj

aii

aaP 11= -

|A|. B . , ijjikk ababjikab === ;;,, . B

nnb

ijb

jibb 11 , P

P. , |A| |B| - ( )nji ,...,,...,,...,1 ( )nij ,...,,...,,...,1 . , (. 1). , - |A| |B| P . - P . || BPP

ppA = == .

. , ( 2.). .

3.1. , .

• 12 . , . - . , =.

AB = . 3. AAAB == . .

6. .

n- . i- j- - .

6. (n-1) , i- j- - aij ij .

4. A :

=

4444

3333

2222

1111

A .

: 23 . 41 .

;

333

333

111

= .444

333

222

23dcbdcbdcb

=

n- n2 .

1.

=

nnnn

n

aaa

aaaa

A

21

22221

11

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

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

00

, nnn

n

aa

aaaaA

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

...

2

222

111111 == .

. , , . naaaaP ...3211= . ( ) ,...,,,1 ( ) ,...,, , 1 - .

( ) ( )1111

,...,,3211

,...,,13211 .. === aaaaaaaaaA nn

.

.

• 13

2.

=

nnnn

ij

n

aaa

a

aaa

A

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

0....................00............................................................................

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

21

11211

, ijijji aA = +)1( .

. A. . : i - (i-1)- , (i-1)- (i-2)- , , 2- 1- . : j- (j-1)-, (j-1)- (j-2)-, , 2- 1-. 3 - :

===

nnnn

n

ij

i

nnnn

ij

n

aaa

aaa

a

aaa

a

aaa

A

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

..........

0................00

)1(

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

0................00...................................

..........

||

21

11211

1

21

11211

nnnn

n

ij

ji

aaa

aaa

a

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

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

0...............00

)1()1(

21

11211

11 = .

1. :

ijijji

nnnn

n

ij a

aaa

aaa

a == ++ )1(

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

......

(-1)|A|

21

11211

j...