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ìï<br />
3 1<br />
x1+ x2 - 2x3<br />
=<br />
2 2<br />
ï<br />
í -x2 - x3<br />
= 2<br />
4x2+ 10x3<br />
=-1<br />
ï<br />
ïî<br />
Êðîê 2.  äðóãîìó ð³âíÿíí³ à 22 = –1≠ 0. ijëèìî éîãî íà<br />
–1. Âèêëþ÷àºìî õ 2 ç óñ³õ ð³âíÿíü, êð³ì äðóãîãî.  ðåçóëüòàò³<br />
ìàºìî<br />
ìï<br />
7 7<br />
x1 - x3<br />
=<br />
2 2<br />
ï<br />
í x2 + x3<br />
=-2<br />
14x3<br />
=-9.<br />
ï<br />
ïî<br />
Êðîê 3. Çâîðîòíèé õ³ä. ²ç òðåòüîãî ð³âíÿííÿ çíàõîäèìî<br />
õ 3 = –9/14, ³ç äðóãîãî — õ 2 = –19/14, ³ç ïåðøîãî — õ 1 = 5/4.<br />
Çàóâàæåííÿ 1. Ïåðåòâîðåííÿ Ãàóññà çðó÷íî ïðîâîäèòè<br />
íå ç ñàìèìè ð³âíÿííÿìè, à ç ðîçøèðåíîþ ìàòðèöåþ, ÿêà<br />
ñêëàäåíà ³ç êîåô³ö³ºíò³â ïðè íåâ³äîìèõ ³ â³ëüíèõ ÷ëåíàõ<br />
äëÿ ð³âíÿííÿ (3.1.1). Ðîçøèðåíà ìàòðèöÿ À * ìຠâèä:<br />
æ2 3 4 1ö<br />
- A* = 2 2 5 3<br />
-<br />
.<br />
çè4 2 2 1<br />
÷ ø<br />
Çðó÷í³ñòü çàñòîñóâàííÿ ðîçøèðåíî¿ ìàòðèö³ ïðîäåìîíñòðóºìî<br />
íà ðîçãëÿíóòîìó ïðèêëàä³<br />
æ 3 1ö æ 7 7ö<br />
1 -2 1 0 -<br />
2 2 2 2<br />
A* ~ 0 -1 - 1 2 ~ 0 1 1 -2 .<br />
0 -4 10 -1 0 0 14 -9<br />
è ç<br />
÷ ø èç<br />
ø<br />
÷<br />
Çâ³äêè íåâàæêî çíàéòè ðîçâ’ÿçîê äàíî¿ ñèñòåìè.<br />
Çàóâàæåííÿ 2. Ïåðåä òèì ÿê ðîçâ’ÿçóâàòè ÑËÀÐ<br />
ÿêèì-íåáóäü ìåòîäîì, äîö³ëüíî çà òåîðåìîþ Êðîíåêåðà–Êàïåëë³<br />
âèçíà÷èòè, ñóì³ñíà âîíà àáî í³. Ïðè ñóì³ñíîñò³ ñèñòåìè<br />
ìîæëèâ³ äâà âèïàäêè: rang À = n àáî rang A < n. Â ïåðøîìó<br />
âèïàäêó ñèñòåìà ìຠºäèíèé ðîçâ’ÿçîê, â äðóãîìó —<br />
áåçë³÷.<br />
Çàóâàæåííÿ 3. Ñë³ä ñêàçàòè, ùî ðîçâ’ÿçóþ÷è ÑËÀÐ<br />
øëÿõîì ïåðåòâîðåííÿ ðîçøèðåíî¿ ìàòðèö³, âèêîíàííÿ óìîâ<br />
òåîðåìè Êðîíåêåðà–Êàïåëë³ ëåãêî âñòàíîâëþºòüñÿ. Öå ùå<br />
ðàç ï³äêðåñëþº åôåêòèâí³ñòü ìåòîäó Ãàóññà. Äî ðå÷³, ó ïðèêëàä³<br />
3.5.1, ÿê öå âèäíî ç ïåðåòâîðåíî¿ ìàòðèö³ A*,<br />
rang A = rang A*.<br />
Ïðèêëàä 3.5.2. Äîâåñòè çà äîïîìîãîþ ïåðåòâîðåíü Ãàóññà,<br />
ùî ñèñòåìà ð³âíÿíü<br />
ì x1+ 2x2- x3<br />
=<br />
7<br />
ï<br />
í2x1- 3x2 + x3<br />
= 3<br />
ï ïî 4x1+ x2- x3<br />
= 5<br />
º íåñóì³ñíîþ.<br />
Ð î ç â ’ ÿ ç à í í ÿ. Ïåðåòâîðèìî ðîçøèðåíó ìàòðèöþ ñèñòåìè<br />
æ1 2 1 7ö æ1 2 1 7ö æ1 2 1 7ö<br />
- - -<br />
2 3 1 3 0 7 3 11 0 7 3 11<br />
- : - - - : - - -<br />
.<br />
çè4 1 -1 5 ÷ ø çè0 0 0 -12 ÷ ø çè0 0 0 -12<br />
÷ ø<br />
Îòæå, ð³âíÿííÿ, ÿêå â³äïîâ³äຠòðåòüîìó ðÿäêó ìàòðèö³, —<br />
ñóïåðå÷ëèâå: âîíî çâåëîñü äî íåâ³ðíî¿ ð³âíîñò³: 0 = –12.<br />
Çâ³äñè âèïëèâàº, ùî äàíà ñèñòåìà íåñóì³ñíà.<br />
Òóò òåîðåìà Êðîíåêåðà–Êàïåëë³ ãàðíî ïðàöþº: rang A =2,<br />
rang A* = 3 (ïåðåâ³ðòå!), ³ îñê³ëüêè 2 < 3, òî äàíà ñèñòåìà<br />
íåñóì³ñíà.<br />
Ïðèêëàä 3.5.3. Ìåòîäîì Ãàóññà ðîçâ’ÿçàòè ñèñòåìó ð³âíÿíü:<br />
ì x1+ x2 + x3<br />
=<br />
5<br />
ï<br />
í2x1-x2- x3<br />
= 4<br />
ï ïî 2x1+ 2x2 + 2x3<br />
= 10<br />
78 79
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