Ture o Cramer me tona tono

ture o Cramer - ko tetahi o nga tikanga tangohia mō te whakaoti ngā pūnaha o whārite taurangi rārangi (Veve rahi). Ko tōna tika tika ki te whakamahi o te tikanga e o te kopu pūnaha, me pera me etahi o nga here i utaina i roto i te tohu o te ture tauwehe.

He pūnaha o ngā whārite taurangi rārangi ki whakarea no ki, hei tauira, he plurality o R - tau tūturu o taurangi x1, x2, ..., xn ko te kohinga o kīanga

ai2 x1 + ai2 x2 + ... ain xn = bi ki i = 1, 2, ..., m, (1)

te wahi aij, bi - tau tūturu. i huaina Ia o enei mau parau e te whārite rārangi, aij - whakarea o te taurangi, bi - whakarea motuhake o whārite.

otinga o (1) lave ki pere n-ahu x ° = (x1 °, x2 °, ..., xn °), i nei whakauru ki te pūnaha mo te x1 taurangi, x2, ..., xn, ia o te rārangi i roto i te pūnaha riro whārite pai .

huaina te pūnaha ko ōrite ki te reira i te iti rawa kia kotahi otinga, me te maiorooro, ki te hāngai te reira ki te huinga otinga o te huinga kau.

Me mahara te reira e i roto i te tikanga ki te kitea rongoā ki ngā pūnaha o whārite rārangi mā te whakamahi i te tikanga o Cramer, ngā pūnaha kopu i ki kia tapawha, e waiwai te tikanga te taua maha o taurangi me whārite i roto i te pūnaha.

Na, ki te whakamahi i tikanga o Cramer, me i te iti rawa mohio koe he aha te Matrix ko te pūnaha o ngā whārite taurangi rārangi, me te i whakaputaina ai. A tuarua, ki matau huaina te mea ko te tokoingoa o te kopu, me ona ake pūkenga o te tätaitanga.

A mana'o na tatou e tenei matauranga riro koe. Whakamiharo! Na ka whai koe ki te tamau aau tika tātai whakatau tikanga Kramer. Hei te faaohie aauraa whakamahi i te momotuhi e whai ake nei:

• Det - te tokoingoa matua o te kopu o te pūnaha;

• deti - Ko te tokoingoa o te kopu whiwhi i te kopu tuatahi o te pūnaha i te whakakapi tīwae i-th o te kopu ki te pere tīwae ko nga taha tika o te whārite taurangi rārangi nei huānga;

• n - te maha o taurangi me whārite i roto i te pūnaha.

Na ture tätaitanga o Cramer i-th xi wae (i = 1, .. n) n-ahu taea te tuhituhia pere x rite

xi = deti / Det, (2).

I roto i tenei take, Det tino rerekē i te kore.

Ko te ahurei o te otinga o te pūnaha, ka te tahi whakaratohia ai e te huru te kore ōritenga o te tokoingoa matua o te pūnaha ki te kore. Kore, ki te mea te moni o (xi), tapawha, tino pai, ka SLAE he infeasible he kopu tapawha. Ka taea e puta tenei i roto i ngā ka i te iti rawa tetahi o kahorekore deti.

Tauira 1. Hei whakaoti i te pūnaha Lau e toru-ahu te whakamahi i tātai o Cramer.
2 x1 + x2 + x3 = 31 4,
5 x1 + x2 + x3 = 2 29,
3 x1 - x2 + x3 = 10.

Whakatau. tuhituhi iho tatou i te kopu o te aho pūnaha i raina, kei hea Hai - ko te rarangi i-th o te kopu.
A1 = (1 2 4), A2 = (5 1 2), A3 = (3, -1, 1).
Tīwae whakarea free b = (31 Oketopa 29).

Ko te pūnaha matua ko te tokoingoa Det
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.

Ki te tātai i te kōwhiringa raupapa det1 te whakamahi i a11 = B1, a21 = b2, a31 = B3. ka
det1 = B1 a22 a33 + a12 a23 B3 + a31 b2 a32 - a13 a22 B3 - B1 a32 a23 - a33 b2 a12 = ... = -81.

Oia atoa, ki whakatatau det2 whakamahi whakauru a12 = B1, a22 = b2, a32 = B3, a, i runga i, ki te tātai i det3 - a13 = B1, a23 = b2, a33 = B3.
Na taea te tirotiro koe e det2 = -108, me det3 = - 135.
E ai ki nga tātai kitea Cramer x1 = -81 / (- 27) = 3, x2 = -108 / (- 27) = 4, x3 = -135 / (- 27) = 5.

Whakahoki kupu: x ° = (3,4,5).

Falala i runga i te faaohiparaa o tenei ture, ka taea te whakamahi i te tikanga o Kramer whakaoti pūnaha o whārite rārangi autaki, hei tauira, ki te tūhura i te pūnaha i runga i te maha ka taea o rongoā i runga i te uara o te k tawhā rānei.

Tauira 2. Hei whakatau i aha ngā uara o te k kore ōritenga tawhā | kx - y - 4 | + | x + ky + 4 | <= 0 kua rite kotahi otinga.

Whakatau.
Tenei ōritenga, i te whakamāramatanga o te mahi kōwae e taea te whakamana anake, ki te he kīanga e rua kore te wā kotahi. Na reira, heke tenei raruraru e ki te kimi i te otinga o ngā whārite taurangi rārangi

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

Ko te otinga ki tenei pūnaha anake, ki te he te reira i te tokoingoa matua o te
Det = k ^ {2} + 1 he kahorekore. Ko reira mārama e he makona hoki uara tūturu katoa o te k tawhā tenei huru.

Whakahoki kupu: hoki ngā uara tūturu katoa o te k tawhā.

taea hoki te iti te whāinga o tenei momo maha ngā raruraru mahi i roto i te mara o te pāngarau, ahupūngao matū ranei.