Cramer ta mulki da kuma ta aikace-aikace

Cramer ta sarautar - shi ne daya daga cikin ainihin hanyoyin warware tsarin na mikakke algebraic lissafai (Slough). Its daidaito saboda da amfani da determinants na tsarin matrix, kazalika da wasu daga cikin hani hõre a cikin gwaji na Theorem.

A tsarin na mikakke algebraic lissafai da coefficients na zuwa, alal misali, a jam'i na R - real lambobi na unknowns x1, x2, ..., xn ne mai tarin maganganu

ai2 x1 + ai2 x2 + ... ain xn = BI tare i = 1, 2, ..., m, (1)

inda aij, guda biyu - real lambobi. Kowace daga cikin wadannan maganganu da aka kira a mikakke lissafi, aij - coefficients na unknowns, guda biyu - m coefficients na lissafai.

bayani na (1) ake magana a n-girma vector x ° = (x1 °, x2 °, ..., xn °), a da canzawa a cikin tsarin domin unknowns x1, x2, ..., xn, kowane daga cikin Lines a cikin tsarin zama mafi kyau lissafi .

A tsarin da aka kira m idan yana da akalla daya bayani, da kuma saba, idan ta yi daidai da bayani sa na fanko sa.

Yana dole ne a tuna da cewa domin ya sami mafita ga tsarin na mikakke lissafai ta amfani da hanyar na Cramer, matrix tsarin da ya zama square, wanda m nufin wannan yawan unknowns da lissafai a cikin tsarin.

Saboda haka, yin amfani da Cramer ta hanyar, dole ne ka akalla san abin da Matrix ne a tsarin na mikakke algebraic lissafai, kuma shi ne bayar. Kuma abu na biyu, ya gane da abin da ake kira da determinant na matrix da kansa basira na ƙidãyar.

Bari mu ɗauka cewa wannan ilimi da ka mallaka. Ban mamaki! To, dole ka kawai haddace dabarbari kayyade Kramer Hanyar. Don rage wuya haddar amfani da wadannan tsarin rubutu:

• Det - babban determinant na matrix da tsarin.

• deti - ne determinant na matrix samu daga farko matrix da tsarin da ya maye gurbin i-th shafi na matrix zuwa shafi vector wanda abubuwa ne dama bangarorin na mikakke algebraic lissafai.

• n - yawan unknowns da lissafai a cikin tsarin.

Sa'an nan Cramer ta sarautar ƙidãyar i-th bangaren Sin Xi (i = 1, .. n) n-girma vector x za a iya rubuta kamar yadda

Xi = deti / det, (2).

A wannan yanayin, det tsananin daban-daban daga sifili.

The musamman da suka bambanta daga cikin bayani na tsarin lokacin da aka hade bayar da rashin daidaito yanayin babban determinant na tsarin sifili. In ba haka ba, idan da Naira Miliyan Xari (Xi), Squared, tsananin kyau, sa'an nan SLAE wani square matrix ne infeasible. Wannan na iya faruwa a musamman a lokacin da a kalla daya daga deti nonzero.

Misali 1. Don shirya da uku-girma lau tsarin yin amfani da Cramer ta dabara.
2 x1 + x2 + X3 = 31 4,
5 x1 + x2 + X3 = 2 29,
3 x1 - x2 + X3 = 10.

Rarrabẽwa. Mun rubuta matrix da tsarin layin da line, inda Ai - ne i-th jere daga cikin matrix.
A1 = (1 2 4), A2 = (5 1 2), A3 = (3, -1, 1).
Column free coefficients b = (31 Oktoba 29).

Babban tsarin ne 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.

Don lissafi da permutation det1 amfani a11 = B1, a21 = B2, a31 = b3. sa'an nan
det1 = B1 a22 a33 + a12 a23 b3 + a31 B2 a32 - a13 a22 b3 - B1 a32 a23 - a33 B2 a12 = ... = -81.

Hakazalika, domin ka lissafta det2 amfani canzawa a12 = B1, a22 = B2, a32 = b3, kuma, daidai da, yin lissafi det3 - a13 = B1, a23 = B2, a33 = b3.
Sa'an nan za ka iya duba da cewa det2 = -108, kuma det3 = - 135.
Bisa ga dabarbari Cramer sami x1 = -81 / (- 27) = 3, x2 = -108 / (- 27) = 4, X3 = -135 / (- 27) = 5.

Amsa: x ° = (3,4,5).

Bisa dogaro da applicability wannan mulkin, da hanyar da Kramer warware tsarin na mikakke lissafai za a iya amfani da a kaikaice, misali, don gudanar da tsarin a kan zai yiwu yawan mafita dangane da darajar da siga k.

Misali 2. Domin sanin abin da dabi'u na siga k rashin daidaito | kx - y - 4 | + | x + ky + 4 | <= 0 yana daidai da daya bayani.

Rarrabẽwa.
Wannan rashin daidaito, da definition na module aiki za a iya yi kawai idan duka biyu maganganu ne sifili lokaci guda. Saboda haka, wannan matsala an rage wa gano da bayani na mikakke algebraic lissafai

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

Maganin wannan tsarin ne kawai idan yana da babban determinant na
Det = k ^ {2} + 1 ne nonzero. A fili yake cewa wannan yanayin ne gamsu ga duk real dabi'u na siga k.

Amsa: ga duk real dabi'u na siga k.

Da manufofin da irin wannan kuma za a iya rage yawa m matsaloli a fagen ilmin lissafi, kimiyyar lissafi, ko sunadarai.