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

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    08-Dec-2016

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  • 506

    1 / /

    : . .

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    2001

  • 2

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

    1 2- 3- .

    . -

    =+=+

    .,

    222

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    cybxa (1)

    (1)

    .22

    11

    =

    baba

    A (2)

    2- . (1) - :

    .2

    1

    =

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    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

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    x = .

    22

    11

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    y =

    3-

    =++=++

    =++

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    1111

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    dzcybxa (5)

    =

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    A , .3

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    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

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    1

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    . , (6) x

    :

    |||| 1

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  • 5

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  • 6 . + - .

    2 4-

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    ,

    211222112221

    1211 aaaaaaaa

    = .

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

    322311332112312213322113312312332211

    333231

    232221

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    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 .

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    naaaP ....21= .

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  • 8 , -

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    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

    =

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    . |A| |A|. -

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

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

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    =

    nnnn

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    n

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    21

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

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    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. , , .

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    n

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    21

    11211

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

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    || = ,

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

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    || = .

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

    nnb

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

    ppA = == .

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  • 12 . , . - . , =.

    AB = . 3. AAAB == . .

    6. .

    n- . i- j- - .

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    4. A :

    =

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    A .

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    ;

    333

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    =

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    2

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    111111 == .

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

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

    .

    .

  • 13

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

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    21

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    , ijijji aA = +)1( .

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

    ===

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    21

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    11 = .

    1. :

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    a == ++ )1(

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

    ......

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    j...

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