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

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506 1 / / : . . . .. 2001 2 . -, . - " " , - . , . " ", - , . , . , - . . . 20 . - . . 1 2- 3- . . - =+=+.,222111cybxacybxa (1) (1) .2211=babaA (2) 2- . (1) - : .21=ccc (3) , (2) (3) (1). x (1) 2b , 1b . 12222111,,bbcybxacybxa=+=+ ( ) .12211221 bcbcxybba = y ( ) .12211221 cacaybaba = , 01221 baba . - x y : 3 12211221bababcbcx= , .12211221babacacay= (4) 1221 baba , (4) - : 1) 21ba , 2) 12ba . - + 1221 baba 2- Det(A)= .|| 21212211 abbababaA == (4) . , c1b2-b1c2 x A1, - : Det(A1)= .|| 212122111 cbbcbcbcA == y A2, - Det(A2)= .|| 212122112 accacacaA == C (4) 11xT y (1) ,22112211bababcbcx = .22112211babacacay = 3- =++=++=++.,,333322221111dzcybxadzcybxadzcybxa (5) =333222111cbacbacbaA , .321=dddd 4 x 1- 3232 bccb , 2- 3113 cbcb , 3- 1221 cbcb . =++ xcbacbacbacbacbacba )( 123312231213132321 .123312231213132321 cbdcbdcbdcbdcbdcbd ++= , x 0. - .123312231213132321123312231213132321cbacbacbacbacbacbacbdcbdcbdcbdcbdcbdx++++= (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) - . =3332221111cbdcbdcbdA , . , (6) x : |||| 1AAx = . y z: |||| 2AAy = , |||| 3AAz = , 5=3332221112cdacdacdaA , =3332221113dbadbadbaA A 2 3- - d. 1. 3- 3006081102210321=++= . 2 n- . . , n- . n- =nnnnnnaaaaaaaaaA212222111211...................................................... . n2 . , . 2 - 3 . : 1) ; 2) - ; 3) +, -. n , 2 ? - . n . . , - , . - , . - (n-1) . - 1. - . n(n-1) 1=n! c . , - . n n! - , 6 . + - . 2 4- 44434241343332312423222114131211aaaaaaaaaaaaaaaa. 1) a11a23a34a43 a12a23a34a41a32 ? - . 2) a12a31a22a44 ? - . 3) a11a22a33a44 a21a43a14a32? , . ? - , 2112221122211211 aaaaaaaa= . . + a11a 22 - (1,2), - a12 a21 - (2,1). 322311332112312213322113312312332211333231232221131211aaaaaaaaaaaaaaaaaaaaaaaaaaa++= . , - . : + - (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), + , - . , - +, , . . nnnnnnaaaaaaaaaA212222111211......................................................|| = . naaaP ....21= . , . , - . - , ( ) ,...,, , - (1,2, ,n). +, ( ) ,...,, - -, - . , 4. - A n n! naaatP ....21)1(= , ( ) ,...,, - (1,2, ,n), t - . = =),..,(,...,21 naaapA 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=nnnnnnaaaaaaaaaA212222111211...................................................... . A. =nnnnnnaaaaaaaaaA212221212111......................................................' . A . 2. . . =nnnnnnaaaaaaaaaA212222111211...................................................... - . |A| |A|. - nnaaaP 2211= , ( )n ,...,, 21 - - 1,2, ,n. |A| |A|. , |A| |A| - . , - . ( )n ,...,, 21 . - ( )n ,...,, 21 - . - bij=aji, =nnnnnnbbbbbbbbbA212222111211......................................................' . 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 ). nnnnjnjjiniinaaaaaaaaaaaaA21212111211.................................................................................|| = , nnnniniijnjjnaaaaaaaaaaaaB21212111211.................................................................................|| = . nnajjaiiaaP 11= - |A|. B . , ijjikk ababjikab === ;;,, . B nnbijbjibb 11 , P P. , |A| |B| - ( )nji ,...,,...,,...,1 ( )nij ,...,,...,,...,1 . , (. 1). , - |A| |B| P . - P . || BPPppA = == . . , ( 2.). . 3.1. , . 12 . , . - . , =. AB = . 3. AAAB == . . 6. . n- . i- j- - . 6. (n-1) , i- j- - aij ij . 4. A : =4444333322221111dcbadcbadcbadcbaA . : 23 . 41 . ;33333311123dbadbadba= .44433322223dcbdcbdcb= n- n2 . 1. =nnnnnaaaaaaaA212222111......................................................00, nnnnaaaaaaA....................2222111111 == . . , , . naaaaP ...3211= . ( ) ,...,,,1 ( ) ,...,, , 1 - . ( ) ( )1111,...,,3211,...,,13211 .. === aaaaaaaaaA nn . . 13 2. =nnnnijnaaaaaaaA........................................................................................0....................00.........................................................................................2111211, ijijji aA = +)1( . . A. . : i - (i-1)- , (i-1)- (i-2)- , , 2- 1- . : j- (j-1)-, (j-1)- (j-2)-, , 2- 1-. 3 - : === nnnnnijinnnnijnaaaaaaaaaaaaaaA............................................................................................................................0................00)1(.............................................................................0................00.............................................||211121112111211nnnnnijjiaaaaaaa.......................................................................................................................................0...............00)1()1(211121111 = . 1. : ijijjinnnnnij aaaaaaaa == ++ )1(........................................................................................................(-1)|A|2111211ji . . 7. ijjiijA = +)1( - ija . 14 , . - , , . - . 4. - : ininiiii AaAaAaA +++= ...2211 . . =+++nnnnniiinniiinaaaaaaxxxaaaaaa2111211211121111211 . = =),..,(,...,21 naaapA P . kx . - , 1x , 2x , , nx . 11xT - 1x ; 22 xT - 2x ; nn xT - nx . nn xTxTxTB +++= ...2211 . 1T . C 11 =x , 02 =x , , 0=nx . 2 11 iAT = . , 22 iAT = , , inn AT = . ninii xAxAxAB +++= ...2211 . innii axaxax === ,...,, 2211 , ininiiii AaAaAaBA +++== ....2211 . . 2 4.1. : 15njnjjjjj AaAaAaA +++= ...2211 . 5. V, , =nnnnvvvvvvV0000 22211211. . V. 4, nnnnnnnnnvvvvvvvvvvvvvV 2211222112221121100000|| ==== . , . . 2. . 4.2. ( ) ( ) . . . 4. BAaAaAa inknikik =+++ ....2211 . B =nnnnknkkknkknaaaaaaaaaaaa21212111211. i k . c 3.1 0. 16 5. ( ). . nnnnininiiiinaaaacacacacacacaaaA21221111211''''''''''''''''''|| +++= i- =++++++= inininiiiiii AacacAacacAacacA )''''''(...)''''''()''''''( 222111 |,''|''|'|')''...''''('')'...''(' 22112211 BcBcAaAaAacAaAaAac ininiiiiininiiii +=+++++++= nnnniniinaaaaaaaaaB212111211'''|'| = , nnnniniinaaaaaaaaaB212111211''''''|''| = . . 3. ( ) , 0 . . . , 5. - ( ), 0 (c 3.1). 5 3 4. , , - . - . . - , .. , (. 5). . 7. . 17 nn - =++++++++++++++++++nnnmnmnmnnnmmmmmmmmmnmmmmmmmmmmmmmmmaaaaaaaaaaaaaaaaaaaaaaaaaaaA212122212222211211111211212211121100000022000 . =mmmmmmaaaaaaaaaB212222111211, =000000000O , = ++++++nmnnmmmmmmmmaaaaaaaaaC212222111211, =++++++++++++nnnmnmnmmmmmnmmmmmaaaaaaaaaD212221212111, A =DCOBA . 6.. - . m. m=1. 11 . 1 . , )11(,1 = nkkm m=k+1. -jjkjjkjjjnn aAaAaAaA 111111111111111 )1(... ==++=++=+= . j1 - a1j , . . j1 1- j- . , 18k, j1 Djj 11 '= , j1' - , B j- . ||||')1(||||')1()1( 111111111111111 BDaDDaaA jjkjjjjkjjjjkjj ====++=++=++= . , m=k+1. 6. =4312712111000320021A . 6, 10)3142)(2221(43123221|| ===A . . - =nnnnnnaaaaaaaaaA212222111211...................................................... . k 19 7 (Laplace P.). n- - k . - k- , k , - . 8. . , n- - n n , , 1. , n n : =+++=+++=+++................................................,,22112222212111212111nnnnnnnnnnbxaxaxabxaxaxabxaxaxa=nnnnnnaaaaaaaaaA212222111211...................................................... , - =nbbbb21 . A nnnnnnaaaaaaaaa212222111211......................................................= . 8 (Cramer G.). 0, 20 , = 11x , = 22x , , = nnx , i - , i- . : 1) , , - ( ); 2) , - . 1) , 1x , 2x , , nx - . - , : =+++=+++=+++................................................,,22112222212111212111nnnnnnnnnnbxaxaxabxaxaxabxaxaxa (10) 1- 11A , 21A , , 1nA . 1x , 2x , , nx . ...)...()...( 1221221112211212111111 ++++++++ nnnn AaAaAaxAaAaAax (11) 12121111212111 ...)...( nnnnnnnn AbAbAbAaAaAax +++=++++ 1121211111 ... nn AaAaAa +++ A ( 4.1). 1121211111 ... nn AaAaAa +++ = . 2x , , nx 0, - ( 4.2). - 1212111 ... nn AbAbAb +++ (11) - nnnnnnaabaabaab2222211211......................................................= 21 . (11) 1x = 1 . 0, = 11x . , 1- 12A , 22A , , 2nA , = 22x . . 1- nA1 , nA2 , , nnA , = nnx . , 0, = 11x , = 22x , , = nnx . 2) , = 11x , = 22x , , = nnx - (10). . i i- bi, - ...({1)(1 2121111112121111212111 ++=+++=+++AbAbaaaaaaa nnnn =++++++++++ )}...()...() 221112222121121 nnnnnnnnnn AbAbAbaAbAbAbaAb ...)...()...({1 2122122111211121211111 ++++++++ nnnnAaAaAabAaAaAab)}...( 1212111 nnnnnn AaAaAab ++++ . , a11A11+a21A21+ an1An1= , a1iA11+a2iA21+ aniAn1=0 (i 1). 11212111 baaann =+++ . , = 11x , = 22x , , = nnx . , , - . . . 22 . 1. .., .. . .: , 1999.-294 . 2. . . . .: , 1975.-400 . 3. .. . .: , 1988.-548 . 4. . . . .: . . 1998.-319 . : , , . .

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