8-§. Sízíqlí teńlemeler sistemasí

Kramer qádesi
Meyli bizge \[n\] belgisizli \[n\] teńlemeler sistemasí berilgen bolsín:

\[\left\{ \begin{align}

& {{a}_{11}}{{x}_{1}}+{{a}_{12}}{{x}_{2}}+...+{{a}_{1s}}{{x}_{s}}+...+{{a}_{1n}}{{x}_{n}}={{b}_{1}}, \\
& {{a}_{21}}{{x}_{1}}+{{a}_{22}}{{x}_{2}}+...+{{a}_{2s}}{{x}_{s}}+...+{{a}_{2n}}{{x}_{n}}={{b}_{2}}, \\
& \,\,.\,\,\,\,\,\,\,.\,\,\,\,\,\,\,\,.\,\,\,\,\,\,\,\,.\,\,\,\,\,\,\,.\,\,\,\,\,\,\,.\,\,\,\,\,\,\,.\,\,\,\,\,\,\,.\,\,\,\,\,\,\,\,.\,\,\,\,\,\,\,\,.\,\,\,\,\,\,\,.\,\,\,\,\,\,\,\,. \\
& {{a}_{n1}}{{x}_{1}}+{{a}_{n2}}{{x}_{2}}+...+{{a}_{ns}}{{x}_{s}}+...+{{a}_{nn}}{{x}_{n}}={{b}_{n}} \\
\end{align} \right.\] (8.2)

Egerde \[{{x}_{i}},\,\,(i=\overline{1,n})\] belgisizlerin olarǵa sáykes \[{{\beta }_{i}},\,\,(i=\overline{1,n})\] sanlar menen almastírǵanda (8.2) teńlemeniń hár biri birdeylikke aylansa, ol jaǵdayda \[{{\beta }_{1}},{{\beta }_{2}},...,\,{{\beta }_{n}}\] sanlar kópligin (8.2) teńlemeler sistemasíníń sheshimi dep ataymíz.

Sistemaníń belgisizleriniń aldíndaǵí koeffitsentlerden tómendegi determinattí dúzeyik:

\[d=\left| \begin{align}

& {{a}_{11}}\,\,\,\,{{a}_{12}}\,...\,\,{{a}_{1s}}\,...\,{{a}_{1n}} \\
& {{a}_{21}}\,\,\,\,{{a}_{22}}\,...\,{{a}_{2s}}\,...\,{{a}_{2n}} \\
& .\,\,\,.\,\,\,\,.\,\,\,\,.\,\,\,.\,\,\,\,\,.\,\,\,\,.\,\,\,\,.\,\,\,\,\,. \\
& {{a}_{n1}}\,\,\,\,{{a}_{n2}}\,...\,{{a}_{ns}}\,...\,{{a}_{nn}} \\
\end{align} \right|\] (8.3)

Buní (8.2) sistemaníń determinantí deymiz. Endi \[d\ne 0\] bolǵanda teńlemeler sistemasíníń sheshiminiń qalay tabílatuǵínín kóreyik. Biz, determinanttíń qaysí bir qatar yamasa baǵanasín almayíq, \[d\] determinant hámme waqítta usí qatar yamasa baǵananíń elementleri menen olardíń sáykes algebralíq tolíqtíríwshílaríníń qosíndísína teń bolatuǵínín bilemiz. Basqasha aytqanda,

\[d={{a}_{i1}}{{A}_{i1}}+{{a}_{i2}}{{A}_{i2}}+...+{{a}_{in}}{{A}_{in}},\,\,\,(i=1,2,...,n)\] (8.4)
bolatuǵínlíǵí túsinikli.
Usí síyaqlí, bazíbir qatar yamasa baǵana elementleri menen basqa qatar yamasa baǵananíń sáykes elementlerine tiyisli algebralíq tolíqtíríwshílardan dúzilgen kóbeymelerdiń qosíndísí nolge teń bolatuǵínlíǵí aníq. Basqa sóz benen aytqanda
\[{{a}_{i1}}{{A}_{j1}}+{{a}_{i2}}{{A}_{j2}}+...+{{a}_{in}}{{A}_{jn}}=0,\,\,\,(i\ne j)\] (8.5)
boladí.
Endi (8.2) sistemaníń teńlemelerin sáykes \[{{A}_{1s}},{{A}_{2s}}...{{A}_{ss}}...{{A}_{ns}}\] lerge kóbeytip keyin teńlemelerdi aǵzama-aǵza qosamíz. Nátiyjede, tómendegige iye bolamíz:

\[\begin{align}

& ({{a}_{11}}{{A}_{1s}}+{{a}_{21}}{{A}_{2s}}+...+{{a}_{n1}}{{A}_{ns}}){{x}_{1}}+({{a}_{12}}{{A}_{1s}}+{{a}_{22}}{{A}_{2s}}+...+{{a}_{n2}}{{A}_{ns}}){{x}_{2}}+... \\
& \\
& ...+.({{a}_{1s}}{{A}_{1s}}+{{a}_{2s}}{{A}_{2s}}+...+{{a}_{ns}}{{A}_{ns}}){{x}_{s}}+...+({{a}_{1n}}{{A}_{1s}}+{{a}_{2n}}{{A}_{2s}}+...+{{a}_{nn}}{{A}_{ns}}){{x}_{n}}= \\
& \\
& ={{b}_{1}}{{A}_{1s}}+{{b}_{2}}{{A}_{2s}}+...+{{b}_{n}}{{A}_{ns}}. \\
\end{align}\]

Biraq, (8.5) formulaǵa muwapíq

\[{{a}_{1k}}{{A}_{1s}}+{{a}_{2k}}{{A}_{2s}}+...+{{a}_{nk}}{{A}_{ns}},\,\,\,\,\,(k\ne s)\]
kórinisindegi hámme qosíndílar teń. Soníń ushín
\[{{a}_{1s}}{{A}_{1s}}+{{a}_{2s}}{{A}_{2s}}+...+{{a}_{ns}}{{A}_{ns}}=d\]
bolíp, bunda basqa hámme skobkalar joǵalíp ((8.4)-formulaǵa qaralsín), nátiyjede biz mínaǵan erisemiz:
\[d\cdot {{x}_{s}}={{b}_{1}}{{A}_{1s}}+{{b}_{2}}{{A}_{2s}}+...+{{b}_{n}}{{A}_{ns}}\] (8.6)
Endi (8.6) teńlemeniń oń tárepi menen shuǵíllanamíz. \[{{a}_{1s}},{{a}_{2s}},...,{{a}_{ns}}\] koeffitsentleriniń ornína saltań aǵzalardí qoyayíq, yaǵníy \[s\]-baǵana menen \[d\] dan paríq qílatuǵín
\[{{d}_{s}}=\left| \begin{align}
& {{a}_{11}}\,\,{{a}_{12}}\,\,...\,{{b}_{1}}\,...\,{{a}_{1n}} \\
& {{a}_{21}}\,\,{{a}_{22}}\,...\,{{b}_{2}}\,...\,{{a}_{2n}} \\
& .\,\,\,.\,\,\,.\,\,\,.\,\,\,.\,\,\,.\,\,\,.\,\,\,.\,\,\,.\,\,\,.\,\,\,. \\
& {{a}_{n1}}\,\,{{a}_{n2}}\,...\,{{b}_{n}}\,...\,{{a}_{nn}} \\
\end{align} \right|\]
determinanttí alayíq hám oní \[s\]-baǵana elementleri boyínsha jaysaq, onda (8.6) teńlemeniń oń tárepi kelip shíǵadí. Demek, (8.6) teńlemeni tómendegishe jazíwǵa boladí:
\[d\cdot {{x}_{s}}={{d}_{s}}\] (8.7)
Egerde \[d\ne 0\] bolsa, onda \[{{x}_{s}}=\frac{{{d}_{s}}}{d}\] \[(s=1,2,...n)\] boladí.
Bunnan
\[{{x}_{1}}=\frac{{{d}_{1}}}{d},\,\,\,\,\,{{x}_{2}}=\frac{{{d}_{2}}}{d},\,\,\,...\,\,\,,\,\,\,\,{{x}_{n}}=\frac{{{d}_{n}}}{d}\] (8.8)

bolatuǵínlíǵí kelip shíǵadí.

Meyli, \[{{\beta }_{1}},{{\beta }_{2}},...,{{\beta }_{n}}\] ler (8.2) teńlemeler sistemasíníń bazíbir sheshimi bolsín. (8.2) teńlemede hár bir \[{{x}_{i}}\] diń ornína sáykes \[{{\beta }_{i}}\] sanlarí menen almastíríp, biz (8.2) sistemaníń hámme teńlemelerin birdeylikke aylandíramíz. (8.7) sistemaní payda etiw ushín (8.2) teńlemeler sistemasí ústinde qanday túrlendiriwler alíp barílǵan bolsa, sońǵí birdeylikler ústinde de sonday túrlendiriwlerdi orínlaymíz. Sózsiz oníń menen biz birdeyliklerdi buzbaǵan bolamíz hám tómendegi nátiyjege erisemiz:
\[d\cdot {{\beta }_{1}}={{d}_{1}},\,\,\,\,\,\,d\cdot {{\beta }_{2}}={{d}_{2}},...,\,d\cdot {{\beta }_{n}}={{d}_{n}}.\]
Solay etip, \[{{\beta }_{1}},{{\beta }_{2}},...,{{\beta }_{n}}\] sanlarí (8.2) sistemaníń sheshimi bolíp tabíladí. (8.8) ańlatpa Kramer formulasí delinedi.
Kramer qádesiniń áhmiyeti, bul qáde qollaníwí múmkin bolǵan jaǵdaylarda sistemaníń sheshimi ushín bul sistemaníń koeffitsentleri aníq ańlatpaní beredi. Biraq, Kramer qádesin praktikada qollaníw kóp ǵana uzínnan-uzaq esaplawlardí talap etedi. \[n\] belgisizli \[n\] teńlemeler sistemasí berilgen bolsa, onda \[n\]-tártipli \[n+1\] determinanttí esaplawǵa tuwra keledi. Joqarída bayan etilgen, belgisizlerdi izbe-iz joǵaltíw usílí bul usíldan anaǵurlím qolaylí bolíp esaplanadí.
8.1-mísal. Teńlemeler sistemasín sheshiń:



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