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Çàóâàæåííÿ. Ïåðåä òèì ÿê çíàõîäèòè ðîçâ’ÿçêè<br />
ÑËÀÐ (3.1.1), áàæàíî äîñë³äèòè ¿¿ íà ðîçâ’ÿçí³ñòü. Ó çâ’ÿçêó<br />
ç öèì ðîçãëÿíåìî äåê³ëüêà íàéïðîñò³øèõ ïðèêëàä³â.<br />
Ïðèêëàä 3.1.1. Äîñë³äèòè ÑËÀÐ âèãëÿäó<br />
⎧x1 + x2<br />
= 5<br />
⎨<br />
⎩x1 + x2<br />
= 0.<br />
Ïåâíà ð³÷, ùî öÿ ñèñòåìà íå ìຠðîçâ’ÿçêó.<br />
Ïðèêëàä 3.1.2. Äîñë³äèòè ÑËÀÐ âèãëÿäó<br />
⎧ x1 + x2<br />
= 1<br />
⎨<br />
2x<br />
+ 2x<br />
= 2<br />
⎩ 1 2 .<br />
Î÷åâèäíî, ùî òàêà ñèñòåìà ìຠíåñê³í÷åííó ìíîæèíó<br />
ðîçâ’ÿçê³â (x 1 = c, x 2 =1–c, äå ñ — áóäü-ÿêå ÷èñëî).<br />
Ïðèêëàä 3.1.3.<br />
⎧x1 + x2<br />
= 2<br />
⎨<br />
⎩x1 − x2<br />
= 0.<br />
Ö³ëêîì î÷åâèäíî, ùî ðîçãëÿäóâàíà ñèñòåìà ìຠºäèíèé<br />
ðîçâ’ÿçîê (x 1 =1, x 2 = 1).<br />
Ðîçãëÿíåìî á³ëüø äåòàëüíî ïðîáëåìè ðîçâ’ÿçíîñò³ ÑËÀÐ.<br />
ÑËÀÐ íàçèâàºòüñÿ ñóì³ñíîþ, ÿêùî âîíà ìຠðîçâ’ÿçîê, ³<br />
íåñóì³ñíîþ, ÿêùî âîíà íå ìຠðîçâ’ÿçêó.  ñâîþ ÷åðãó ñóì³ñíà<br />
ñèñòåìà ìîæå áóòè âèçíà÷åíîþ (ìຠºäèíèé ðîçâ’ÿçîê)<br />
³ íåâèçíà÷åíîþ (ìຠíå ºäèíèé ðîçâ’ÿçîê). Ìàòðèöÿ À êîåô³ö³ºíò³â<br />
ïðè íåâ³äîìèõ ó ñèñòåì³ (3.1.1) íàçèâàºòüñÿ îñíîâíîþ.<br />
Ïðèºäíóþ÷è äî ìàòðèö³ À ñòîâïåöü â³ëüíèõ ÷ëåí³â<br />
ñèñòåìè (3.1.1), îòðèìàºìî òàê çâàíó ðîçøèðåíó ìàòðèöþ<br />
À* äàíî¿ ìàòðèö³<br />
A*<br />
⎛a11 a12 ... a1n<br />
b1<br />
⎞<br />
⎜<br />
⎟<br />
⎜<br />
a a ... a b<br />
⎟<br />
... ... ... ... ... .<br />
⎜<br />
am 1<br />
am2<br />
... amn b ⎟<br />
⎝<br />
m ⎠<br />
21 22 2n<br />
2<br />
= ⎜ ⎟<br />
3.2. ÑÈÑÒÅÌÀ n ˲ͲÉÍÈÕ ÀËÃÅÁÐÀ¯×ÍÈÕ<br />
вÂÍßÍÜ Ç n ÍÅ<strong>²</strong>ÄÎÌÈÌÈ<br />
Íåõàé ÷èñëî ð³âíÿíü ñèñòåìè (3.1.1) äîð³âíþº ÷èñëó íåâ³äîìèõ,<br />
òîáòî m = n. Òîä³ ìàòðèöÿ ñèñòåìè (3.1.3) º êâàäðàòíîþ,<br />
à ¿¿ âèçíà÷íèê ⏐A⏐ íàçèâàºòüñÿ âèçíà÷íèêîì ö³º¿<br />
ñèñòåìè.<br />
3.2.1. Ñèñòåìà äâîõ ð³âíÿíü ç äâîìà íåâ³äîìèìè<br />
Ðîçãëÿíåìî ñèñòåìó äâîõ àëãåáðà¿÷íèõ ð³âíÿíü ç äâîìà<br />
íåâ³äîìèìè<br />
⎧a11x1 + a12x2 = b1<br />
;<br />
⎨<br />
⎩a21x1 + a22x2 = b2<br />
,<br />
(3.2.1)<br />
â ÿê³é õî÷à á îäèí êîåô³ö³ºíò ïðè íåâ³äîìèõ â³äì³ííèé â³ä<br />
íóëÿ. Äëÿ ðîçâ’ÿçêó ö³º¿ ñèñòåìè âèêëþ÷èìî íåâ³äîìó x 2 ,<br />
ïîìíîæèâøè ïåðøå ð³âíÿííÿ íà à 22 , äðóãå — íà (–à 12 ), ³ äîäàìî<br />
çì³íåí³ ð³âíÿííÿ. Ïîò³ì âèêëþ÷èìî íåâ³äîìó x 1 , ïîìíîæèâøè<br />
ïåðøå ð³âíÿííÿ íà (–à 21 ), äðóãå — íà (à 11 ), ³ òàêîæ<br />
äîäàìî çì³íåí³ ð³âíÿííÿ.  ðåçóëüòàò³ îòðèìàºìî ñèñòåìó<br />
⎧( a11a22 − a21a12 ) x1 = ba<br />
1 22<br />
−b2 a12,<br />
⎨<br />
⎩( a11a22 − a21a12 ) x2 = b2a11 −ba<br />
1 21<br />
.<br />
(3.2.2)<br />
Âèðàç â äóæêàõ ñèñòåìè (3.2.2) º íå ùî ³íøå, ÿê âèçíà÷íèê<br />
ñèñòåìè (3.2.1)<br />
∆= 11 12<br />
a11a<br />
− 22<br />
a21a<br />
= 12<br />
a21 a<br />
.<br />
22<br />
Ââåäåìî òàêîæ òàê³ ïîçíà÷åííÿ<br />
b a a b<br />
∆ = − = ∆ = − = .<br />
1 12 11 1<br />
1<br />
ba<br />
1 22<br />
ba<br />
2 12<br />
,<br />
2<br />
ba<br />
2 11<br />
ba<br />
1 21<br />
b2 a22 a21 b2<br />
Òîä³ ñèñòåìó (3.2.2) ìîæíà çàïèñàòè ó âèãëÿä³:<br />
⎧∆⋅ x1 =∆1;<br />
⎨<br />
⎩ ∆⋅ x2 =∆<br />
2.<br />
a<br />
a<br />
(3.2.3)<br />
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