Mokhoa oa Cramer le kopo ea oona

Tsela ea Cramer ke e 'ngoe ea mekhoa e nepahetseng ea ho rarolla mekhoa ea li-equge algebraic equations (SLAE). Ho nepahala ha eona ho bakoa ke tšebeliso ea lihlopha tsa matrix ea tsamaiso, hammoho le lithibelo tse itseng tse behiloeng nakong ea bopaki ba theorem.

Tsamaiso ea likaroloana tse tloaelehileng tsa algebraic le li-coefficients, ka mohlala, ho sethaleng sa linomoro tsa R-sebele, ho tloha ho tse sa tsejoeng x1, x2, ..., xn ke sete sa lipolelo tsa foromo

Ai2 x1 + ai2 x2 + ... ain xn = bi bakeng sa i = 1, 2, ..., m, (1)

Moo aij, bi ke linomoro tsa sebele. E 'ngoe le e' ngoe ea lipolelo tsena e bitsoa equation e lekanang, aij - coefficients bakeng sa sa tsejoeng, li-coefficients tse sa lefelloeng tsa equation.

Tharollo ea tsamaiso (1) ke vector n-dimension x ° = (x1 °, x2 °, ..., xn °), e leng ha e kenngoa tsamaisong, ho e-na le tse sa tsejoeng x1, x2, ..., xn, e mong le o mong oa mela ea tsamaiso e fetoha tekanyo ea 'nete .

Ho boleloa hore tsamaiso e kopane haeba e na le tharollo e le 'ngoe' me e sa lumellane haeba tharollo ea eona e lumellana le setei se se nang letho.

E tlameha ho hopoloa hore e le hore o fumane tharollo ho mekhoa ea li-equation algebraic linear, sebelisa mokhoa oa Cramer, matrices a tsamaiso e lokela ho ba lisekoere, e leng se bolelang palo e le 'ngoe ea tse sa tsejoeng le ho lekanngoa tsamaisong.

Kahoo, e le hore u sebelise mokhoa oa Cramer, e tlameha ebe motho o lokela ho tseba hore na matrix a mekhoa ea li-equation algebraic e lekaneng le hore na e ngotsoe joang. 'Me ka lekhetlo la bobeli, ho utloisisa se bitsoang se khethiloeng sa matrix le ho tseba tsebo ea ho bala.

Nka hore u na le tsebo ena. E makatsang! Joale o tlameha ho hopola mekhoa e khethollang mokhoa oa Cramer. Ho nolofatsa ho hopola, re sebelisa lintlha tse latelang:

• Det ke sona se ka sehloohong se hlahisang matrix;

• Diti ke eona e ikhethileng ea matrix e fumanoang ho tloha matrix ea tsamaiso haeba karolo ea i-th ea matrix e nkeloa sebaka ke khohlopo ea khohlopo eo likarolo tsa eona li leng lehlakoreng le letona la litsamaiso tsa li-equation algebraic linear;

• N ke palo ea tse sa tsejoeng le li-equation tsamaisong ena.

Ka nako eo Phula ea Cramer e busa bakeng sa ho sebelisa khomphutha ea i-th xi (i = 1, ... n) ea n-dimensional vector x e ka ngoloa ka foromo

Khothalletsa hore u bala Xi = sethaleng / Det, (2).

Det ke tieo feela.

E ikhethang ea tharollo ea tsamaiso ha e lumellana e netefatsa hore mohloli o ka sehloohong oa tsamaiso ke zero. Ho seng joalo, haeba tjhelete (xi), squared, e tiile hantle, joale SLAE e nang le matrix ea square e tla ba e sa lumellaneng. Sena se ka etsahala, haholo-holo, ha bonyane e 'ngoe ea li-date e fapane le zero.

Mohlala 1 . Lokisa tsamaiso ea litekanyetso tse tharo tsa LAU ho sebelisa litlhahlobo tsa Cramer.
X1 + 2 x2 + 4 x3 = 31,
5 x1 + x2 + 2 x3 = 29,
3 x1 - x2 + x3 = 10.

Tharollo. Re ngola sekhetho sa tsamaiso ea moeli ka mohala, moo Ai e leng lehlakoreng la matrix.
A1 = (1 2 4), A2 = (5 1 2), A3 = (3 -1 1 1).
Karolo ea li-coefficients tse sa lefelloeng b = (31 29 10).

Tsela e ka sehloohong ea Det system ke
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.

E le hore re bale det1, re sebelisa sebaka sa a11 = b1, a21 = b2, a31 = b3. Joale
Det1 = b1 a22 a33 + a12 a23 b3 + a31 b2 a32 - a13 a22 b3 - b1 a32 a23 - a33 b2 a12 = ... = -81.

Ka tsela e ts'oanang, ho bala det2, re sebelisa phetoho e 12 = b1, a22 = b2, a32 = b3 mme, ka ho le joalo, ho bala det3 - a13 = b1, a23 = b2, a33 = b3.
Joale o ka hlahloba hore det2 = -108, le det3 = -135.
Ho ea ka litlhahlobo tsa Cramer, re fumana x1 = -81 / (-27) = 3, x2 = -108 / (-27) = 4, x3 = -135 / (-27) = 5.

Karabo ke: x ° = (3,4,5).

Ho itšetlehile ka maemo a ho sebelisoa ha molao ona, mokhoa oa Cramer oa ho rarolla mekhoa ea li-equation e lekanang e ka sebelisoa ka tsela e sa tobang, ka mohlala, ho batlisisa tsamaiso bakeng sa mekhoa e mengata ea tharollo ho latela boleng ba parameter e k.

Mohlala 2. Hlahloba hore na ke litekanyetso life tsa parameter k ea ho se lekane | kx - y - 4 | + | + + ky + 4 | <= 0 e na le tharollo e le 'ngoe feela.

Tharollo.
Ho se lekane hona, ka lebaka la tlhaloso ea mokhoa oa ts'ebetso, ho ka khotsofatsoa feela haeba lipolelo tseo ka bobeli li lekanang le zero. Ka lebaka leo, bothata bona bo fokotsa ho fumana tharollo ea tsamaiso e lumellanang ea algebraic equations

Kx - y = 4,
X + ky = -4.

Tharollo ea tsamaiso ena e ikhethile haeba e ka sehloohong e ikhethileng
Det = k ^ {2} + 1 ha e na zero. Ho totobetse hore boemo bona bo khotsofatsoa ke litekanyetso tsohle tsa sebele tsa parameter k.

Karabo: bakeng sa litekanyetso tsohle tsa sebele tsa parameter k.

Ho mathata a mofuta ona, mathata a mangata a sebetsang a tsoang tšimong ea lipalo, filosofi kapa k'hemistri a ka boela a fokotseha .