Kutonga Cramer uye kushandiswa kwayo

Kutonga Cramer kuti - mumwe chaiwo nzira kuti nekugadzirisa mugadziriro Linear algebraic equations (Slough). Kururama Its nokuda kushandiswa determinants wegadziriro chizvaro, uyewo mamwe pezvaunorambidzwa iri uchapupu theorem no.

Tsika Linear algebraic equations pamwe coefficients enzvimbo Somuenzaniso, imwe muzvizhinji kuti R - chaicho nhamba unknowns x1, x2, ..., xn muunganidzwa mashoko

ai2 x1 + ai2 x2 + ... Aini xn = Bi pamwe ini = 1, 2, ..., m, (1)

apo aij, bi - chaiyo nhamba. Mumwe nemumwe mashoko aya anonzi ari Linear mukana wokuti vanhu vanga, aij - coefficients of unknowns, bi - akazvimiririra coefficients pakati equations.

mhinduro (1) raireva N-dimensional Vector X ° = (x1 °, x2 °, ..., xn °), panguva iyo kuzvitsinhanisa mugadziriro nokuda unknowns x1, x2, ..., xn, mumwe nomumwe mitsetse munyika kunova yakanaka mukana wokuti vanhu vanga .

The Hurongwa unonzi kupindirana kana ane zvinenge imwe mhinduro, uye dzisingapindirani, kana rinobatana nemhinduro yakatarwa yaJehovha chinhu yakatarwa.

Zvinofanira kurangarirwa kuti kuitira kuwana zvinogadzirisa mugadziriro Linear equations kushandisa nzira Cramer, chizvaro hurongwa kuva akaenzana, iro rinoreva zvakafanana uwandu unknowns uye equations munyika.

Saka, kushandisa Cramer kuti nzira, iwe unofanira runenge kuziva zvakaita Matrix ndiyo kwegadziriro Linear algebraic equations, uye rakapa. Uye chechipiri, kunzwisisa chii inonzi determinant pakati chizvaro uye yayo unyanzvi computation.

Ngationei kufunga kuti ruzivo urwu hwaunahwo. Anoshamisa! Zvino unofanira chete nomusoro Masvomhu kusarudza Kramer nzira. Kurerutsa nomusoro kushandisa anotevera notation:

• Det - huru determinant pakati chizvaro wegadziriro;

• deti - ndiye determinant pakati chizvaro yakawanikwa kubva chikuru chizvaro wegadziriro nokutsiva i-fung mbiru chizvaro kuti wechikamu Vector vane zvinhu zviri panhivi rworudyi Linear algebraic equations;

• N - uwandu unknowns uye equations munyika.

Ipapo Cramer kutonga computation i-fung zvinoriumba XI (i = 1, .. n) N-dimensional Vector X anogona kunyorwa se

XI = deti / Det, (2).

Muchiitiko chino, Det kuomerera zvakasiyana razero.

Kusiyana nemhinduro wegadziriro kana anorenda inopiwa ne kusaruramisira ezvinhu chikuru determinant wegadziriro kusvikira razero. Zvikasadaro, kana mubatanidzwa (XI), akaenzana, chaizvoizvo akanaka, zvino SLAE sikweya chizvaro iri infeasible. Izvi zvinogona kuitika zvikurukuru apo pane mumwe deti nonzero.

Muenzaniso 1. Kugadzirisa-dimensional matatu Lau maitiro achishandisa Cramer kuti mutoo.
2 x1 + x2 + x3 = 31 4,
5 x1 + x2 + x3 = 2 29,
3 x1 - x2 + x3 = 10.

Decision. Tinonyora pasi chizvaro wegadziriro mutsetse norwonzi, apo Ai - ndiwo i-fung musara chizvaro.
A1 = (1 2 4), A2 = (5 1 2), A3 = (3, -1, 1).
Column vakasununguka coefficients b = (31 October 29).

The chikuru Hurongwa ndiyo 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.

Kuverenga permutation det1 kushandisa A11 = B1, a21 = b2, A31 = B3. ndokubva
det1 = B1 a22 a33 + a12 a23 B3 + A31 b2 a32 - a13 a22 B3 - B1 a32 a23 - a33 b2 a12 = ... = -81.

Saizvozvowo, kuti compute det2 kushandisa kuzvitsinhanisa a12 = B1, a22 = b2, a32 = B3, uye, naizvozvovo, kuverenga det3 - a13 = B1, a23 = b2, a33 = B3.
Zvadaro unogona kutarisa kuti det2 = -108, uye det3 = - 135.
Maererano Masvomhu Cramer vanowana x1 = -81 / (- 27) = 3, x2 = -108 / (- 27) = 4, x3 = -135 / (- 27) = 5.

Pindura: x ° = (3,4,5).

Kuvimba applicability utongi ichi, kuti nzira Kramer kugadzirisa mugadziriro Linear equations zvingashandiswa zvisina kunanga, Somuenzaniso, kuongorora hurongwa pamusoro zvaibvira nhamba zvinogadzirisa zvichienderana kukosha parameter k.

Muenzaniso 2. Kuti uone pane tsika yacho parameter, k kusaruramisira | Kx - Y - 4 | + | X + KY + 4 | <= 0 ane chaizvo imwe mhinduro.

Decision.
kusaruramisira uku, kubudikidza tsanangudzo module basa kunogona kuitwa chete kana vaviri mashoko vari razero panguva imwe chete. Naizvozvo, dambudziko iri ipfupi dzokuwana mhinduro kubva Linear algebraic equations

Kx - Y = 4,
X + KY = -4.

The dambudziko ino chete kana chikuru determinant of
Det = k ^ {2} + 1 riri nonzero. Zviri pachena kuti ichi ndicho kugutsikana vose chaicho tsika yacho parameter k.

Kupindura nokuti zvose chaicho tsika yacho parameter k.

The zvebasa ichi mhando kunogona kuderedzwa matambudziko akawanda anobatsira mune masvomhu, wefizikisi kana kemisitiri.