Ulamuliro Cramer ndi tanthauzo lake

ulamuliro Cramer a - ndi imodzi mwa njira yolondola kuthetsa machitidwe a maikwezhoni liniya algebraic (Slough). molondola chifukwa cha ntchito ya determinants a dongosolo masanjidwewo, komanso ena mwa zinthu woyikidwa mu umboni wa Lingaliro Lovomerezeka ndi.

dongosolo la maikwezhoni liniya algebraic ndi koefishienti za thupi, mwachitsanzo, ndi kuchuluka kwa R - manambala weniweni wa unknowns x1, x2, ..., xn ndi mndandanda wa mawu

ai2 x1 + ai2 x2 + ... Ain xn = zina kuti adzipeza ndi ine = 1, 2, ..., m, (1)

kumene aij, zina kuti adzipeza - manambala enieni. Aliyense wa mawu amenewa amatchedwa ndi aone kuti pali liniya, aij - koefishienti wa unknowns ndi zina kuti adzipeza - koefishienti osadalira pa maikwezhoni.

yankho la (1) anatchula N-azithunzi omwe tikunena vekitala × ° = (x1 °, x2 °, ..., xn °), limene M'MALO mu dongosolo la unknowns x1, x2, ..., xn, aliyense wa mizere mu dongosolo anakhala aone kuti pali .

Kachitidwe amatchedwa zogwirizana ngati ali njira imodzi, ndi kosagwirizana, ngati chikugwirizana ndi akonzedwa njira ya kuika kanthu.

Kumbukirani kuti kupeza njira kachitidwe wa maikwezhoni liniya ntchito njira Cramer, kachitidwe masanjidwewo kukhala lalikulu, lomwe limatanthauza zochuluka mofana unknowns ndi maikwezhoni mu dongosolo.

Choncho, ntchito njira Cramer, muyenera osachepera mukudziwa zimene masanjidwewo ndi dongosolo la maikwezhoni liniya algebraic, ndipo inaika. Lachiwiri, kumvetsa otchedwa determinant ya masanjidwewo ndi luso yakeyake masovedwe.

Tiyeni tinene kuti chidziwitso cha izi muli nacho. Zodabwitsa! Ndiye muli kuloweza chabe mitunduyi kudziwa Kramer njira. Wosalira kuloweza ntchito kalembedwe zotsatirazi:

• Det - ndi determinant yaikulu ya masanjidwewo a dongosolo;

• deti - ndi determinant ya masanjidwewo kuwapeza masanjidwewo chachikulu cha dongosolo ndi kuchotsa i-TH ndime ya masanjidwewo ku vekitala ndime amene zinthu ndi mbali lamanja la maikwezhoni liniya algebraic;

• N - chiwerengero cha unknowns ndi maikwezhoni mu dongosolo.

Ndiye Cramer ulamuliro masovedwe i-TH chigawo XI (i = 1, .. n) N-azithunzi omwe tikunena vekitala × akhoza kulembedwa ngati

XI = deti / Det, (2).

Pankhaniyi, Det mosamalitsa wosiyana ziro.

Wapadera njira ya dongosolo pamene ichitikire operekedwa ndi chikhalidwe lochitira wa determinant chachikulu cha dongosolo kuti ziro. Apo ayi, ngati Uwerenge (XI), lofanana mbali zonse, mosamalitsa, ndiye SLAE bokosi lalikulu ndi infeasible. Izi angayambe makamaka pamene mmodzi wa nonzero deti.

Chitsanzo 1. Pofuna kuthana ndi atatu azithunzi omwe tikunena lau dongosolo ntchito chilinganizo Cramer a.
2 x1 + x2 + x3 = 31 4,
5 x1 + x2 + x3 = 2 29,
3 x1 - x2 + x3 = 10.

Kusankha. Ife kulemba masanjidwewo a dongosolo mzere ndi mzere, kumene Ai - ndi mzere i-TH a masanjidwewo lapansi.
A1 = (1 2 4), A2 = (5 1 2), A3 = (3, -1 1).
Ndime ufulu koefishienti B = (31 October 29).

Dongosolo ndicho determinant 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.

Kuwerengetsa permutation det1 ntchito a11 = B1, B2 a21 =, a31 = B3. ndiye
det1 = B1 a22 a33 + a12 a23 B3 + a31 B2 a32 - a13 a22 B3 - B1 a32 a23 - a33 B2 a12 = ... = -81.

Mofananamo, kuti kugwira ntchito zowerengetsa det2 ntchito M'MALO a12 = B1, B2 a22 =, a32 = B3, ndi moyenera, kuwerengera det3 - a13 = B1, B2 a23 =, a33 = B3.
Ndiye inu mukhoza onetsetsani kuti det2 = -108, ndi det3 = - 135.
Malinga mitunduyi ndi Cramer kupeza x1 = -81 / (- 27) = 3, x2 = -108 / (- 27) = 4, x3 = -135 / (- 27) = 5.

Yankho: × ° = (3,4,5).

Modalira applicability ya lamulo limeneli, njira Kramer kuthetsa machitidwe a maikwezhoni liniya angagwiritsidwe ntchito ayi, mwachitsanzo, kufufuza dongosolo pa nambala zotheka njira malingana phindu la chizindikiro K.

2. Pofuna kudziwa zimene mfundo za chizindikiro K lochitira | Kx - Y - 4 | + | × + KY + 4 | <= 0 ali chimodzimodzi njira imodzi.

Kusankha.
lochitira ino, mwa tanthauzo la ntchito gawo angathe kuchitidwa kokha ngati mawu onse ziro imodzi. Choncho, vuto ili yafupika kupeza yankho la maikwezhoni liniya algebraic

Kx - Y = 4,
× + KY = -4.

Yankho dongosolo kokha ngati ndi determinant yaikulu ya
Det = K ^ {2} + 1 nonzero. Zikuonekeratu kuti chikhalidwe ichi amakhutitsidwa kuti mfundo zonse weniweni wa chizindikiro K.

Yankho: pakuti mfundo zonse weniweni wa chizindikiro K.

Zolinga za mtundu uwu akhoza yafupika mavuto zambiri m'munda wa masamu, sayansi kapena zimapangidwira.