Xeerka Cramer iyo codsiga

xeerka Cramer ee - waa mid ka mid ah hababka saxda ah ee xalilaada nidaamyada isleegyada aljabra toosan (Slough). Its saxnimada ay sabab u tahay isticmaalka leh marxaladaha of furta nidaamka, iyo sidoo kale qaar ka mid ah xayiraad lagu soo rogay in caddaynta ah ee Aragtida ah.

Nidaamka A of isleegyada aljabra leh horgalaha ay leeyihiin, tusaale ahaan, hannaanka ah R - tirada dhabta ah ee wax dahsoon x1, x2, ..., xn waa a collection of tibaaxaha

ai2 x1 + x2 ai2 + ... Cayn xn = bi la i = 1, 2, ..., m, (1)

halkaas oo aij, bi - tirada dhabta ah. Mid kasta oo ka erayadan waxaa loo yaqaan isla'egyada toosan, aij - horgalaha ah dahsoon ee, bi - horgalaha madax banaan isla'egyada.

xal of (1) gudbiyo n-cabbir dulinka x ° = (x1 °, x2 °, ..., xn °), kaasoo la bedelay galay nidaamka loogu talagalay x1 ka dahsoon, x2, ..., xn, mid kasta oo ka mid ah khadadka ee nidaamka noqdo isla'eg ugu fiican .

Nidaamka waxa loo yaqaan joogto haddii waxa uu leeyahay ugu yaraan hal xal, oo aan joogto ahayn, haddii ay ku beegan tahay set xal of go'an madhan.

Waa in la xusuusnaado in si si xal loogu nidaamka isleegyo qumman oo isticmaalaya habka ay u Cramer heli, nidaamyada furta leeyihiin in ay laba jibbaaran, kaas oo asal ahaan ka dhigan tahay tiro la mid ah dahsoon iyo isleegyo nidaamka.

Sidaas daraaddeed, in la isticmaalo habka Cramer ee, waa in aad ugu yaraan ogaan waxa Matrix waa nidaam isleegyada aljabra toosan, oo waxaa la soo saaray. Iyo marka labaad, si ay u fahmaan waxa la yiraahdo ahayn tilmaame oo ka mid ah shaxda iyo xirfado u gaar ah xisaabinta.

Ina keena aynu ka fikirno in aqoonta this aad hantidaan. Wonderful! Markaas waxaad u leedahay in kaliya uu korka qaaciidooyinka go'aaminayo habka Kramer. Si loo fududeeyo xasuusta isticmaali sal soo socda:

• Det - ahayn tilmaame ugu weyn ee maxalka furta oo ka mid ah nidaamka;

• deti - Waa ahayn tilmaame ee maxalka furta oo ka furta aasaasiga ah ee nidaamka helay bedelay column i-aad ee maxalka furta oo si dulinka column kuwaas oo xubno ka yihiin dhinacyada saxda ah ee isleegyada aljabra toosan a;

• n - tirada dahsoon iyo isleegyo nidaamka.

Markaas Cramer ee xeerka xisaabinta i-th xi qayb (i = 1, .. n) n-cabbir dulinka x waxaa loo qori karaa sida

xi = deti / Det, (2).

Xaaladdan oo kale, Det si adag uga duwan eber.

aqonsado ee xalka nidaamka marka ay si wadajir ah waxaa bixiya xaalad sinnaan ah ahayn tilmaame ugu muhiimsan ee nidaamka eber ah. Haddii kale, haddii wadarta (xi), afar gees, si adag u wanaagsan, ka dibna SLAE shax square waa infeasible. Tani waxay dhici kartaa, gaar ahaan marka ay ugu yaraan mid ka mid ah nonzero deti.

Tusaale 1. Si loo xaliyo nidaamka Lau saddex dhinac-cabbir iyadoo la isticmaalayo formula Cramer ee.
2 + x1 x2 + x3 = 31 4,
5 x1 + x2 + x3 = 2 29,
3 x1 - x2 + x3 = 10.

Go'aan. Waxaan ku qoro furta oo ka mid ah line nidaamka by line, halkaas oo Aacii - waa isku xigta i-aad ee maxalka furta oo ah.
A1 = (1 2 4), A2 = (5 1 2), A3 = (3, -1, 1).
Column horgalaha free b = (31 October 29).

Nidaamka ugu weyn waa Det ka ahayn tilmaame
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.

Si loo xisaabiyo permutation ku det1 isticmaalaya A11 = B1, B2 = a21, a31 = B3. ka dibna
det1 = B1 a22 a33 + A12 a23 B3 + a31 a32 B2 - a13 a22 B3 - B1 a32 a23 - a33 B2 = A12 ... = -81.

Sidoo kale, xisaabinta isticmaalka bedelay det2 A12 = B1, B2 = a22, a32 = B3, iyo, sida, si ay u xisaabiso det3 - a13 = B1, B2 = a23, a33 = B3.
Markaas waxaad eegi kartaa det2 in = -108, iyo det3 = - 135.
Sida laga soo xigtay qaaciidooyinka Cramer heli x1 = -81 / (- 27) = 3, x2 = -108 / (- 27) = 4, x3 = -135 / (- 27) = 5.

Jawaab: x ° = (3,4,5).

Ku tiirsan applicability ee qaanuunkan, habka of Kramer xalinta nidaamka isleegyo qumman loo isticmaali karaa si dadban, tusaale ahaan, si ay u baaraan nidaamka tirada suurto galka ah ee xal ku xiran tahay qiimaha k dhimaya ah.

Tusaale 2. Si loo go'aamiyo waxa qiyamka sinnaan dhimaya k ah | kx - y - 4 | + | x + ky + 4 | <= 0 sax ah mid xal leeyahay.

Go'aan.
sinnaan Tani, by qeexidda function module ah lagu samayn karaa oo keliya haddii labada tibaaxaha waa eber isku mar. Sidaa darteed, dhibaatadan waa la dhimay si loo helo xalka of isleegyada aljabra toosan

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

Xalka in nidaamka this kaliya haddii ay tahay ahayn tilmaame ugu weyn ee
Det = k ^ {2} + 1 waa nonzero. Waxaa cad in xaaladda this qanacsan ee dhammaan qiimaha dhabta ah ee k dhimaya yahay.

Jawaab: waayo, dhammaan qiimaha dhabta ah ee k dhimaya ka.

Ujeedooyinka noocan ah sidoo kale waa la dhimi karaa dhibaatooyin badan oo wax ku ool ah oo duurka ku ah xisaabta, fisikiska ama kiimikada.