Cramer tus txoj cai thiab nws daim ntawv thov

Cramer tus txoj cai - yog ib tug ntawm lub caij nyoog txoj kev daws systems ntawm linear algebraic equations (Slough). Nws cov neeg vim txoj kev siv cov determinants ntawm cov system matrix, raws li zoo raws li ib co ntawm cov kev txwv yuam nyob rau hauv lub pov thawj ntawm cov theorem.

Ib tug system ntawm linear algebraic equations nrog coefficients teej tug mus rau, piv txwv li, ib tug plurality ntawm R - cov zauv tiag tiag ntawm unknowns x1, x2, ..., xn yog ib tug sau ntawm kab zauv

ai2 x1 + ai2 x2 + ... ain xn = lis nrog kuv = 1, 2, ..., m, (1)

qhov twg aij, lis - cov zauv tiag tiag. Txhua yam ntawm cov kab zauv yog hu ua ib tug linear kab zauv, aij - coefficients ntawm lub unknowns, lis - ywj siab coefficients sib npaug.

tov ntawm (1) xa mus rau n-dimensional vector x ° = (x1 °, x2 °, ..., xn °), uas hloov mus rau hauv lub system rau lub unknowns x1, x2, ..., xn, txhua tus kab nyob rau hauv lub system yuav zoo tshaj plaws kab zauv .

Lub system no yog hu ua raws li yog hais tias nws muaj tsawg kawg yog ib tug tov, thiab inconsistent, yog hais tias nws coincides nrog cov tshuaj set ntawm cov empty set.

Nws yuav tsum tau nco ntsoov tias twb nyob rau hauv thiaj li yuav nrhiav kev daws rau nruab ntawm linear equations siv cov qauv ntawm cov Cramer, matrix tshuab yuav tsum tau square, uas yeej txhais tau tias tib tus naj npawb ntawm unknowns thiab sib npaug nyob rau hauv lub system.

Yog li ntawd, siv Cramer txoj kev, koj yuav tsum yog tsawg kawg yog paub dab tsi lub Matrix yog ib tug system ntawm linear algebraic equations, thiab nws yog muab. Thiab secondly, to taub li cas yog hu ua tus determinant ntawm tus matrix thiab nws tus kheej cov txuj ci ntawm le caag.

Cia peb xav hais tias txoj kev paub no koj muaj. Zoo kawg nkaus! Ces koj yuav tsum cia li cim qauv txiav txim Kramer txoj kev. Mus piv rau kev cim siv cov nram qab no cim:

• Nws - lub ntsiab determinant ntawm tus matrix ntawm lub system;

• deti - yog lub determinant ntawm tus matrix tau los ntawm tus thawj matrix ntawm lub system los ntawm hloov i-th sab ntawm lub matrix rau ib kem vector uas nws hais yog txoj cai sab ntawm linear algebraic equations;

• n - lub xov tooj ntawm unknowns thiab sib npaug nyob rau hauv lub system.

Ces Cramer kav le caag i-th tivthaiv xi (i = 1, .. n) n-dimensional vector x yuav sau raws li

xi = deti / Nws, (2).

Nyob rau hauv cov ntaub ntawv no, Nws nruj me ntsis txawv ntawm xoom.

Lub uniqueness ntawm kev daws ntawm lub system thaum nws yog koom tes ua ke muab los ntawm lub inequality mob ntawm lub ntsiab determinant ntawm lub system rau zero. Txwv tsis pub, yog hais tias tus sum ntawm (xi), squared, nruj me ntsis zoo, ces SLAE ib tug square matrix yog infeasible. Qhov no yuav tshwm sim nyob rau hauv particular thaum tsawg kawg yog ib tug ntawm deti nonzero.

Piv txwv li 1. Yuav kom daws tau peb-dimensional lau system siv Cramer lub mis.
2 x1 + x2 + x 3 = 31 4,
5 x1 + x2 + x 3 = 2 29,
3 x1 - x2 + x 3 = 10.

Kev txiav txim siab. Peb sau cov matrix ntawm lub system kab los ntawm kab, qhov chaw uas Ai - yog i-th kab ntawm lub matrix.
A1 = (1 2 4), A2 = (5 1 2), A3 = (3, -1, 1).
Kem free coefficients b = (31 Lub kaum hli ntuj 29).

Lub ntsiab system yog lub determinant Nws
Nws = 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.

Los mus laij lub permutation det1 siv a11 = B1, a21 = B2, a31 = B3. ces
det1 = B1 a22 a33 + a12 a23 B3 + a31 B2 a32 - a13 a22 B3 - B1 a32 a23 - a33 B2 a12 = ... = -81.

Ib yam li ntawd, mus laij det2 siv hloov a12 = B1, a22 = B2, a32 = B3, thiab, raws li, los xam det3 - a13 = B1, a23 = B2, a33 = B3.
Ces koj yuav xyuas tias det2 = -108, thiab det3 = - 135.
Raws li cov qauv Cramer nrhiav x1 = -81 / (- 27) = 3, x2 = -108 / (- 27) = 4, x 3 = -135 / (- 27) = 5.

Teb: x ° = (3,4,5).

Kev cia siab rau lub applicability ntawm txoj cai no, cov qauv ntawm cov Kramer daws systems ntawm linear equations yuav siv tau kov, piv txwv li, mus soj ntsuam lub kaw lus rau cov tau tus naj npawb ntawm cov ntsiab nyob rau ntawm tus nqi ntawm ib tug parameter k.

Piv txwv li 2. Yuav kom paub dab tsi tseem ceeb ntawm cov parameter k inequality | kx - y - 4 | + | x + ky + 4 | <= 0 muaj raws nraim ib cov tshuaj.

Kev txiav txim siab.
Qhov no tsis sib xws, los ntawm lub ntsiab txhais ntawm lub module muaj nuj nqi yuav tsum tau tsuas yog ob kab zauv uas zero ib txhij. Yog li ntawd, qhov teeb meem no yog txo mus nrhiav kev daws ntawm linear equations algebraic

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

Cov tshuaj rau qhov no system tsuas yog hais tias nws yog lub ntsiab determinant ntawm lub
Nws = k ^ {2} + 1 yog nonzero. Nws yog tseeb hais tias tus mob no yog txaus siab rau rau tag nrho cov tiag tiag qhov tseem ceeb ntawm cov parameter k.

Teb: rau tag nrho cov tiag tiag qhov tseem ceeb ntawm cov parameter k.

Lub hom phiaj ntawm no hom yuav kuj yuav raug txo ntau tswv yim teeb meem nyob rau hauv lub tshav pob ntawm kev kawm txog zauv, physics los yog Science News for KIDS.