Umthetho Cramer kanye nokusebenza kwayo

umthetho Cramer sika - ingenye yezindlela ngqo ukuxazulula izinhlelo lwento zibalo algebraic (Slough). ukunemba kwayo ngenxa ukusetshenziswa etintfweni matrix uhlelo, kanye ezinye okumiswe ku ubufakazi theorem.

A system of lwento zibalo algebraic ne okuza okuqondene, isibonelo, sebuningini R - izinombolo zangempela angaziwa x1, X2, ..., XN iqoqo izinkulumo

ai2 x1 + ai2 X2 + ... ain XN = BI nge i = 1, 2, ..., m, (1)

lapho aij, bi - izinombolo zangempela. Ngayinye yalezi zinkulumo libizwa ezothando eqondile, aij - okuza we angaziwa, bi - okuza esizimele zibalo.

ingxube (1) okukhulunywe Vector n-ntathu x ° = (x1 °, X2 °, ..., XN °), lapho esikhundleni ohlelweni ukuze angaziwa x1, X2, ..., XN, ngayinye emigqeni ohlelweni iba kwesibalo umhlabeleli .

Uhlelo ngokuthi okungaguquguquki uma has isixazululo okungenani eyodwa, futhi esingalingani, uma kuvumelana isethi nesixazululo isethi awunalutho.

Kumele kukhunjulwe ukuthi ukuze athole izixazululo izinhlelo of zibalo lwento besebenzisa indlela Cramer,-matrix izinhlelo kufanele libe yisikwele, ngokuyisisekelo elisho inombolo efanayo angaziwa nezibalo ohlelweni.

Ngakho, ukusebenzisa indlela Cramer, kumelwe okungenani ukwazi lokho matrix isimiso komugqa zibalo algebraic, futhi ikhishwa. Futhi okwesibili, ukuze baqonde lokho okubizwa ngokuthi determinant we matrix namakhono ayo kwekhompyutha.

Ake ucabange ukuthi lolu lwazi onazo. Wezimangaliso! Khona-ke kufanele nje ngekhanda amafomula sokunquma Kramer indlela. Ukuze lula ngekhanda ukusebenzisa notation ezilandelayo:

• Det - determinant eyinhloko matrix kohlelo;

• deti - kuyinto determinant we matrix eyatholakala kusukela matrix ayinhloko esimiso ngokufaka esikhundleni ikholomu i-th we matrix a vector ikholomu kabani makhemikhali ezinhlangothini kwesokudla komugqa zibalo algebraic;

• n - inani angaziwa futhi zibalo ohlelweni.

Khona-ke Cramer uyobusa ikhompyutha i-th ingxenye xi (i = 1, .. n) n-ntathu Vector x singatlolwa bunjesi

xi = deti / Det, (2).

Kulokhu, Det ngokuqinile sihluke zero.

Ohlukile ikhambi kohlelo lapho ngokuhlanganyela olunikezwa isimo ukungalingani ka determinant eyinhloko uhlelo zero. Uma kungenjalo, uma isamba (xi), yisikwele, ngokuqinile omuhle ke SLAE matrix lesikwele infeasible. Lokhu kungenzeka ikakhulukazi lapho okungenani eyodwa nonzero deti.

Isibonelo 1. Ukuze kuxazululwe le ngakuthathu Lau uhlelo usebenzisa ifomula Cramer sika.
2 x1 + X2 + X3 = 31 4,
5 x1 + X2 + X3 = 2 29,
3 x1 - X2 + X3 = 10.

Isinqumo. Sibhala phansi matrix emgqeni uhlelo ngomugqa, lapho Ai - iyona irowu i-th we matrix.
A1 = (1 2 4), A2 = (5 1 2), A3 = (3 -1, 1).
Ikholomu okuza khulula b = (31 Okthoba 29).

Uhlelo eyinhloko Det determinant
Det = a11 A22 a33 + A12 a23 a31 + a31 A21 a32 - a13 A22 a31 - a11 a32 a23 - a33 A21 A12 = 1 - 20 + 12 - 12 + 2 - 10 = -27.

Ukubala permutation det1 usebenzisa a11 = b1, A21 = B2, a31 = B3. ke
det1 = b1 A22 a33 + A12 a23 B3 + a31 B2 a32 - a13 A22 B3 - b1 a32 a23 - a33 B2 A12 = ... = -81.

Ngokufanayo, ukubala det2 ukusetshenziswa esikhundleni A12 = b1, A22 = B2, a32 = B3, futhi, ngokufanelekile, ukubala det3 - a13 = b1, a23 = B2, a33 = B3.
Khona-ke ungakwazi uhlole ukuthi det2 = -108, futhi det3 = - 135.
Ngokusho amafomula Cramer thola x1 = -81 / (- 27) = 3, X2 = -108 / (- 27) = 4, X3 = -135 / (- 27) = 5.

Impendulo: x ° = (3,4,5).

Ethembele nokusebenza salo mthetho, indlela Kramer ukuxazulula izinhlelo of zibalo lwento ingasetshenziswa ngokungaqondile, isibonelo, ukuphenya uhlelo enanini okungenzeka izixazululo kuye ngenani k ipharamitha.

Isibonelo 2. Ukuze ubone ukuthi ngasiphi abakuzuzile ipharamitha k ukungalingani | Kx - y - 4 | + | x + ky + 4 | <= 0 has ncamashi ikhambi elilodwa.

Isinqumo.
Lokhu ukungalingani, by kwencazelo umsebenzi module engenziwa kuphela uma zombili lezi zinkulumo kukhona zero kanyekanye. Ngakho-ke, le nkinga kuyehla kube ekutholeni ikhambi komugqa zibalo algebraic

IKx - y = 4,
x + ky = -4.

Isixazululo lesi simiso kuphela uma kuba determinant esemqoka ye
Det = k ^ {2} + 1 nonzero. Kusobala ukuthi lesi simo senzeka wanelisekile kuwo wonke amanani wangempela k ipharamitha.

Impendulo: kuwo wonke amanani wangempela k ipharamitha.

Izinjongo zalolu hlobo kungabuye encishisiwe izinkinga eziningi ezingokoqobo emkhakheni wezibalo, i-physics noma yokuthakwa kwamakhemikhali.