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Теория Матриц и Определителей

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N?aaeiyy oeiea ( 45.

Ai?iae Iineaa.

O/aiee 10 eeanna “A” Ai?ioia Aaaaiee

Eo?niaay ?aaioa (/a?iiaee).

Aaaaeaiea a oai?eth iao?eoe e ii?aaeaeeoaeae.

1996 aiae.

Iaeaaeaiea.

TOC \o “1-3” Iaeaaeaiea. GOTOBUTTON _Toc375578081 PAGEREF
_Toc375578081 2

1. Iao?eoeu. GOTOBUTTON _Toc375578082 PAGEREF _Toc375578082 3

1.1 Iiiyoea iao?eoeu. GOTOBUTTON _Toc375578083 PAGEREF
_Toc375578083 3

1.2 Iiiaiua iia?aoeee iaae iao?eoeaie. GOTOBUTTON _Toc375578084
PAGEREF _Toc375578084 3

2. Ii?aaeaeeoaee. GOTOBUTTON _Toc375578085 PAGEREF _Toc375578085 5

2.1 Iiiyoea ii?aaeaeeoaey. GOTOBUTTON _Toc375578088 PAGEREF
_Toc375578088 5

2.2 Au/eneaiea ii?aaeaeeoaeae. GOTOBUTTON _Toc375578089 PAGEREF
_Toc375578089 5

2.3 Iniiaiua naienoaa ii?aaeaeeoaeae. GOTOBUTTON _Toc375578090
PAGEREF _Toc375578090 6

3. Nenoaiu eeiaeiuo o?aaiaiee. GOTOBUTTON _Toc375578091 PAGEREF
_Toc375578091 8

3.1 Iniiaiua ii?aaeaeaiey. GOTOBUTTON _Toc375578093 PAGEREF
_Toc375578093 8

3.2 Oneiaea niaianoiinoe nenoai eeiaeiuo o?aaiaiee. GOTOBUTTON
_Toc375578094 PAGEREF _Toc375578094 8

3.3 ?aoaiea neoai eeiaeiuo o?aaiaiee iaoaaeii E?aia?a. GOTOBUTTON
_Toc375578095 PAGEREF _Toc375578095 8

3.4 ?aoaiea neoai eeiaeiuo o?aaiaiee iaoaaeii Aaonna. GOTOBUTTON
_Toc375578096 PAGEREF _Toc375578096 9

4. Ia?aoiay iao?eoea. GOTOBUTTON _Toc375578097 PAGEREF
_Toc375578097 9

4.1 Iiiyoea ia?aoiie iao?eoeu. GOTOBUTTON _Toc375578098 PAGEREF
_Toc375578098 9

4.2 Au/aneaiea ia?aoiie iao?eoeu. GOTOBUTTON _Toc375578099 PAGEREF
_Toc375578099 9

Nienie eeoa?aoo?u. GOTOBUTTON _Toc375578100 PAGEREF _Toc375578100
9

1. Iao?eoeu.

1.1 Iiiyoea iao?eoeu.

Iao?eoeae iacuaaaony i?yiioaieueiay oaaeeoea ec /enae, niaea?aeauay
iaeioi?ia eiee/anoai m no?ie e iaeioi?ia eiee/anoai n noieaoeia.
*enea m e n iacuaathony ii?yaeeaie iao?eoeu. A neo/aa, anee m = n ,
iao?eoea iacuaaaony eaaae?aoiie, a /enei m = n — aa ii?yaeeii.

1.2 Iniiaiua iia?aoeee iaae iao?eoeaie.

Iniiaiuie a?eoiaoe/aneeie iia?aoeeyie iaae iao?eoeaie yaeythony
oiiiaeaiea iao?eoeu ia /enei, neiaeaiea e oiiiaeaiea iao?eoe.

I?aaeaea anaai aeiaiai?einy n/eoaoue iao?eoeu ?aaiuie, anee yoe iao?eoeu
eiatho iaeeiaeiaua ii?yaeee e ana eo niioaaonoaothuea yeaiaiou
niaiaaeatho.

Ia?aeaeai e ii?aaeaeaieth iniiaiuo iia?aoeee iaae iao?eoeaie.

Neiaeaiea iao?eoe: Noiiie aeaoo iao?eoe, iai?eia?: A e B, eiathueo
iaeeiaeiaia eiee/anoai no?ie e noieaoeia, eiuie neiaaie, iaeieo e oao
aea ii?yaeeia m e n iacuaaaony iao?eoea N = ( Nij )( i = 1, 2, …m; j =
1, 2, …n ) oao aea ii?yaeeia m e n, yeaiaiou Cij eioi?ie ?aaiu.

Cij = Aij + Bij ( i = 1, 2, …, m; j = 1, 2, …, n )
( 1.2 )

Aeey iaicia/aiey noiiu aeaoo iao?eoe eniieuecoaony caienue C = A + B.
Iia?aoeey ninoaaeaiey noiiu iao?eoe iacuaaaony eo neiaeaieai

Eoae ii ii?aaeaeaieth eiaai :

=

Ec ii?aaeaeaiey noiiu iao?eoe, a oi/iaa ec oi?ioeu ( 1.2 )
iaiin?aaenoaaiii auoaeaao, /oi iia?aoeey neiaeaiey iao?eoe iaeaaeaao
oaie aea naienoaaie, /oi e iia?aoeey neiaeaiey aauanoaaiiuo /enae, a
eiaiii :

ia?aianoeoaeueiui naienoaii : A + B = B + A

ni/aoaoaeueiui naienoaii : (A + B) + C = A + (B + C)

Yoe naienoaa iicaieytho ia caaioeoueny i ii?yaeea neaaeiaaiey neaaaaiuo
iao?eoe i?e neiaeaiee aeaoo eee aieueoaai /enea iao?eoe.

Oiiiaeaiea iao?eoeu ia /enei :

iacuaaaony iao?eoea C = (Cij) ( i = 1, 2, … , m; j = 1, 2, …, n ),
yeaiaiou eioi?ie ?aaiu

Aij ( i = 1, 2, …, m; j = 1, 2, …, n ). (1.3)

. Iia?aoeey ninoaaeaiey i?iecaaaeaiey iao?eoeu ia /enei iacuaaaony
oiiiaeaieai iao?eoeu ia yoi /enei.

Iaiin?aaenoaaiii ec oi?ioeu (1.3) ynii, /oi oiiiaeaiea iao?eoeu ia /enei
iaeaaeaao neaaeothueie naienoaaie :

?ani?aaeaeeoaeueiui naienoaii ioiineoaeueii noiiu iao?eoe:

B

ni/aoaoaeueiui naienoaii ioiineoaeueii /eneiaiai iiiaeeoaey:

A)

?ani?aaeaeeoaeueiui naienoaii ioiineoaeueii noiiu /enae :

A.

Caia/aiea : ?aciinoueth aeaoo iao?eoe A e B iaeeiaeiauo ii?yaeeia
anoanoaaiii iacaaoue oaeoth iao?eoeo C oao aea ii?yaeeia, eioi?ay a
noiia n iao?eoeae B aeaao iao?eoeo A. Aeey iaicia/aiey ?aciinoe aeaoo
iao?eoe eniieuecoaony anoanoaaiiay caienue : C = A – B.

Ia?aiiiaeaiea iao?eoe :

I?iecaaaeaieai iao?eoeu A = (Aij) ( i = 1, 2, …, m; j = 1, 2, …, n ),
eiathuae ii?yaeee niioaaonoaaiii ?aaiua m e n, ia iao?eoeo B = (Bij) ( i
= 1, 2, …, n;

j = 1, 2, …, p ), eiathuoth ii?yaeee niioaaonoaaiii ?aaiua n e p,
iacuaaaony iao?eoea C = (Nij) ( i = 1, 2, … , m; j = 1, 2, … , p ),
eiathuay ii?yaeee, niioaaonoaaiii ?aaiua m e p, e yeaiaiou Cij,
ii?aaeaeyaiua oi?ioeie

( i = 1, 2, …, m; j = 1, 2, …, p ) (1.4)

Aeey iaicia/aiey i?iecaaaeaiey iao?eoeu A ia iao?eoeo B eniieuecotho
caienue

C = AB. Iia?aoeey ninoaaeaiey i?iecaaaeaiey iao?eoeu A ia iao?eoeo B
iacuaaaony ia?aiiiaeaieai yoeo iao?eoe. Ec noi?ioee?iaaiiiai auoa
ii?aaeaeaiey auoaeaao, /oi iao?eoeo A iiaeii oiiiaeeoue ia ia anyeoth
iao?eoeo B : iaiaoiaeeii /oiau /enei noieaoeia iao?eoeu A auei ?aaii
/eneo no?ie iao?eoeu B. Aeey oiai /oiau iaa i?iecaaaeaiey AB e BA ia
oieueei auee ii?aaeaeaiu, ii e eiaee iaeeiaeiaue ii?yaeie, iaiaoiaeeii e
aeinoaoi/ii, /oiau iaa iao?eoeu A e B auee eaaae?aoiuie iao?eoeaie
iaeiiai e oiai aea ii?yaeea.

Oi?ioea (1.4) i?aaenoaaeyao niaie i?aaeei ninoaaeaiey yeaiaioia iao?eoeu
C,

yaeythuaeny i?iecaaaeaieai iao?eoeu A ia iao?eoeo B. Yoi i?aaeei iiaeii
noi?ioee?iaaoue e neiaanii : Yeaiaio Cij, noiyuee ia ia?ana/aiee i-e
no?iee e j-ai noieaoea iao?eoeu C = AB, ?aaai noiia iiia?iuo
i?iecaaaeaiee niioaaonoaothueo yeaiaioia i-e no?iee iao?eoeu A e j-ai
noieaoea iao?eoeu B. A ea/anoaa i?eia?a i?eiaiaiey oeacaiiiai i?aaeea
i?eaaaeai oi?ioeo ia?aiiiaeaiey eaaae?aoiuo iao?eoe aoi?iai ii?yaeea

Ec oi?ioeu (1.4) auoaeatho neaaeothuea naienoaa i?iecaaaeaiey iao?eoeu A
ia iao?eoeo B :

ni/aoaoaeueiia naienoai : (AB) C = A (BC);

?ani?aaeaeeoaeueiia ioiineoaeueii noiiu iao?eoe naienoai :

(A + B) C = AC + BC eee A (B + C) = AB + AC.

Aii?in i ia?anoaiiai/iii naienoaa i?iecaaaeaiey iao?eoe eiaao niune
noaaeoue eeoue aeey eaaae?aoiuo iao?eoe iaeeiaeiaiai ii?yaeea.
Yeaiaioa?iua i?eia?u iieacuaatho, /oi i?iecaaaeaiee aeaoo eaaae?aoiuo
iao?eoe iaeeiaeiaiai ii?yaeea ia iaeaaeaao, aiiaua aiai?y,
ia?anoaiiai/iui naienoaii. A naiii aeaea, anee iieiaeeoue

Oa aea iao?eoeu, aeey i?iecaaaeaiey eioi?uo ni?aaaaeeeai ia?anoaiaai/iia
naienoai, i?eiyoi iacuaaoue eiiiooe?othueie.

N?aaee eaaae?aoiuo iao?eoe auaeaeei eeann oae iacuaaaiuo aeeaaiiaeueiuo
iao?eoe, o eaaeaeie ec eioi?uo yeaiaiou, ?aniieiaeaiiua aia aeaaiie
aeeaaiiaee, ?aaiu ioeth. N?aaee anao aeeaaiiaeueiuo iao?eoe n
niaiaaeathueie yeaiaioaie ia aeaaiie aeeaaiiaee iniai aaaeioth ?ieue
ea?atho aeaa iao?eoeu. Ia?aay ec yoeo iao?eoe iieo/aaony, eiaaea ana
yeaiaiou aeaaiie aeeaaiiaee ?aaiu aaeeieoea, iacuaaaony aaeeie/iie
iao?eoeae n-iai ii?yaeea e iaicia/aaony neiaieii E . Aoi?ay iao?eoea
iieo/aaony i?e anao yeaiaioao ?aaiuo ioeth e iacuaaaony ioeaaie
iao?eoeae n-iai ii?yaeea e iaicia/aaony neiaieii O. Aeiionoei, /oi
nouanoaoao i?iecaieueiay iao?eoea A, oiaaea

AE = EA = A, AO = OA = O.

Ia?aay ec oi?ioe oa?aeoa?ecoao iniaoth ?ieue aaeeie/iie iao?eoeu A,
aiaeiae/ioth oi ?iee, eioi?oth ea?aao /enei 1 i?e ia?aiiiaeaiee
aauanoaaiiuo /enae. *oi aea eanaaony iniaie ?iee ioeaaie iao?eoeu I, oi
aa auyaeyao ia oieueei aoi?ay ec oi?ioe, ii e yeaiaioa?ii i?iaa?yaiia
?aaainoai : A + O = O + A = A. Iiiyoea ioeaaie iao?eoeu iiaeii
aaiaeeoue e ia aeey eaaae?aoiuo iao?eoe.

2. Ii?aaeaeeoaee.

2.1 Iiiyoea ii?aaeaeeoaey.

I?aaeaea anaai iaiaoiaeeii caiiiieoue, /oi ii?aaeaeeoaee nouanoaotho
oieueei aeey iao?eoe eaaae?aoiiai aeaea, eai aeey iao?eoe ae?oaiai oeia
ia nouanoaoao ii?aaeaeeoaeae. A oai?ee nenoai eeiaeiuo o?aaiaiee e a
iaeioi?uo ae?oaeo aii?inao oaeiaii eniieueciaaoue iiiyoea ii?aaeaeeoaey,
eee aeaoa?ieiaioa.

2.2 Au/eneaiea ii?aaeaeeoaeae.

. Ii?aaeaeeoaeai neaaeothuae iao?eoeu yaeyaony oaeia au?aaeaiea :
a11a22a33 + a12a23a31 + a13a21a32 – a11a23a32 – a12a21a33 – a13a22a31..
Eae au aeaeeoa ii i?in/eouaaaony aeiaieueii eaaei, anee caiiiieoue
ii?aaeaeaiioth iineaaeiaaoaeueiinoue. N iieiaeeoaeueiui ciaeii eaeoo
aeaaiay aeeaaiiaeue e ia?acothueany ec yeaiaioia o?aoaieueieee, eiathuea
ia?aeeaeueioth aeaaiie aeeaaiiaee noi?iio, a aeaiiii neo/aa yoi
o?aoaieueieee a12a23a31, a13a21a32.

N io?eoeaoaeueiui ciaeii eaeoo iiai/iay aeeaaiiaeue e o?aoaieueieee ae
ia?aeeaeueiua, o.a. a11a23a32 , a12a21a33. Oaeei ia?acii iaoiaeyony
ii?aaeaeeoaee ethaiai ii?yaeea. Ii auaatho neo/ae, eiaaea e yoio iaoiae
noaiiaeony aeiaieueii neiaeiui, iai?eia?, eiaaea yeaiaioia a iao?eoea
i/aiue iiiai, e aeey oiai, /oiau nin/eoaoue ii?aaeaeeoaeue ioaeii
cao?aoeoue oeio a?aiaie e aieiaiey.

.

Oai?aia 1.1 Eaeia au ie aue iiia? no?iee i ( i =1, 2 …, n ), aeey
ii?aaeaeeoaey n-iai ii?yaeea ni?aaaaeeeaa oi?ioea

iacuaaaiay ?aceiaeaieai yoiai ii?aaeaeeoaey ii i-e no?iea. Iiae/a?eiai,
/oi a yoie oi?ioea iieacaoaeue noaiaie, a eioi?oth aicaiaeeony /enei
(-1), ?aaai noiia iiia?ia no?iee e noieaoea, ia ia?ana/aiee eioi?uo
noieo yeaiaio Aij.

Oai?aia 1.2 Eaeia au ie aue iiia? noieaoea j ( j =1, 2 …, n ), aeey
ii?aaeaeeoaey n-ai ii?yaeea ni?aaaaeeeaa oi?ioea

iacuaaaiay ?aceiaeaieai yoiai ii?aaeaeeoaey ii j-iio noieaoeo.

2.3 Iniiaiua naienoaa ii?aaeaeeoaeae.

O ii?aaeaeeoaeae oaeaea anoue naienoaa, n iiiiuueth eioi?uo caaea/a eo
au/eneaiey noaiiaeony aieaa eaaeie. Eoae, ieaea onoaiaaeeaaaony ?yae
naienoa, eioi?uie iaeaaeaao i?iecaieueiue ii?aaeaeeoaeue n-ai ii?yaeea.

.

.

. Naienoai aioeneiiao?ee i?e ia?anoaiiaea aeaoo no?ie ( eee aeaoo
noieaoeia ). I?e ia?anoaiiaea ianoaie aeaoo no?ie ( eee aeaoo noieaoeia
) ii?aaeaeeoaeue nio?aiyao naith aaniethoioth aaee/eio, ii iaiyao ciae
ia i?ioeaiiieiaeiue. Aeey ii?aaeaeeoaey aoi?iai ii?yaeea yoi naienoai
i?iaa?yaony yeaiaioa?ii ( ec oi?ioeu au/eneaiey ii?aaeaeeoaey aoi?iai
ii?yaeea n?aco auoaeaao, /oi ii?aaeaeeoaee ioee/athony eeoue ciaeii ).

, aaea

.

Yoe o?e naienoaa yaeythony iniiaiuie naienoaaie ii?aaeaeeoaey,
ane?uaathueie aai i?e?iaeo. Neaaeothuea iyoue naienoa yaeythony
eiae/aneeie neaaenoaeyie o?ao iniiaiuo naienoa.

Neaaenoaea 1. Ii?aaeaeeoaeue n aeaoiy iaeeiaeiauie no?ieaie ( eee
noieaoeaie ) ?aaai ioeth.

Neaaenoaea 2. Oiiiaeaiea anao yeaiaioia iaeioi?ie no?iee ( eee
iaeioi?iai noieaoea ) ii?aaeaeeoaey ia /enei a ?aaiineeueii oiiiaeaieth
ii?aaeaeeoaey ia yoi /enei a. Eiuie neiaaie , iauee iiiaeeoaeue anao
yeaiaioia iaeioi?ie no?iee ( eee iaeioi?iai noieaoea ) ii?aaeaeeoaey
iiaeii auianoe ca ciae yoiai ii?aaeaeeoaey.

Neaaenoaea 3. Anee ana yeaiaiou iaeioi?ie no?iee ( eee iaeioi?iai
noieaoea ) ?aaiu ioeth, oi e nai ii?aaeaeeoaeue ?aaai ioeth.

Neaaenoaea 4. Anee yeaiaiou aeaoo no?ie ( eee aeaoo noieaoeia )
ii?aaeaeeoaey i?iii?oeeiiaeueiu, oi ii?aaeaeeoaeue ?aaai ioeth.

, oi aaee/eia ii?aaeaeeoaey ia eciaiyaony. Neaaenoaea 5, eae e eeiaeiia
naienoai, aeiioneaao aieaa iauoth oi?ioee?iaeo, eioi?oth y i?eaaaeo aeey
no?ie : anee e yeaiaioai iaeioi?ie no?iee ii?aaeaeeoaey i?eaaaeoue
niioaaonoaothuea yeaiaiou no?iee, yaeythuaeny eeiaeiie eiiaeiaoeeae
ianeieueeeo ae?oaeo no?ie yoiai ii?aaeaeeoaey ( n eaeeie oaiaeii
eiyooeoeeaioaie ), oi aaee/aia ii?aaeaeeoaey ia eciaieony. Neaaenoaea 5
oe?iei i?eiaiyaony i?e eiie?aoiii au/eneaiee ii?aaeaeeoaeae.

3. Nenoaiu eeiaeiuo o?aaiaiee.

3.1 Iniiaiua ii?aaeaeaiey.

…….

3.2 Oneiaea niaianoiinoe nenoai eeiaeiuo o?aaiaiee.

…….

3.3 ?aoaiea nenoai eeiaeiuo o?aaiaiee iaoiaeii E?aia?a.

Ecaanoii, /oi eniieuecoy iao?eoeu iu iiaeai ?aoaoue ?acee/iua nenoaiu
o?aaiaiee, i?e /ai yoe nenoaiu iiaoo auoue eaeie oaiaeii aaee/aiu e
eiaoue neieueei oaiaeii ia?aiaiiuo. N iiiiuueth ianeieueeeo auaiaeia e
oi?ioe ?aoaiea ia?iiiuo nenoai o?aaiaiee noaiiaeony aeiaieueii auno?ui e
aieaa eaaeei.

A /anoiinoe, y iieoo iaoiaeu E?aia?a e Aaonna. Iaeeaa/aeoei niiniaii
yaeyaony iaoiae E?aia?a ( aeey iaiy ), eee eae aai aua iacuaatho –
oi?ioea E?aia?a. Eoae, aeiionoei, /oi iu eiaai eaeoth-eeai nenoaio
o?aaiaiee

, aaea ioaaou aoaeoo o?aaiaiee aoaeoo iaoiaeeony a iineaaeiai
noieaoea. Oaia?ue iu aaaaeai iiiyoea iniiaiiai ii?aaeaeeoaey; a aeaiiii
neo/aa ii aoaeao auaeyaeaoue oaeei ia?acii :

. Iniiaiui ii?aaeaeeoaeai eae au oaea caiaoeee yaeyaony iao?eoea
ninoaaeaiiay ec eiyooeoeeaioia noiyueo i?e ia?aiaiiuo. Iie oaeaea eaeoo
a ii?yaeea noieaoeia, o. a. a ia?aii noieaoea noiyo eiyooeoeeaiou,
eioi?ua iaoiaeyony i?e x, ai aoi?ii noieaoea i?e y, e oae aeaeaa. Yoi
i/aiue aaaeii, eai a neaaeothueo aeaenoaeyo iu aoaeai caiaiyoue eaaeaeue
noieaaoe eiyooeoeeaioia i?e ia?aiaiiie ia noieaaoe ioaaoia o?aaiaiee.
Eoae, eae y oaea aiai?ee, iu caiaiyai noieaaoe i?e ia?aie ia?aiaiiie ia
noieaaoe ioaaoia, caoai i?e aoi?ie, eiia/ii yoi ana caaeneo io oiai,
neieueei ia?aiaiiuo iai ioaeii iaeoe.

.

3 . Eae iaoiaeeony ii?aaeaeeoaeue o?aoueaai ii?yaeea au oaea ciaaoa. A
aio caeanue iu e i?eiaiyai i?aaeei E?aia?a. Iii auaeyaeeo oae :

. Ii?aaeaeeoaeue ninoaaeaiiue ec eiyooeoeeaioia i?e iaecaanoiuo,
iacuaaaony ii?aaeaeeoaeai nenoaiu.

3.4 ?aoaiea nenoai eeiaeiuo o?aaiaiee iaoiaeii Aaonna.

…….

4. Ia?aoiay iao?eoea.

4.1 Iiiyoea ia?aoiie iao?eoeu.

……

4.2 Au/eneaiea ia?aoiie iao?eoeu.

……

Nienie eeoa?aoo?u.

A. A. Eeueei, Y. A. Iiciye “Eeiaeiay Aeaaa?a”

2. A. Ae. Eei, A. A. Oeeei “Yeaiaioa?iua i?aia?aciaaiey a eeiaeiie
aeaaa?a”

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