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Ïðèêëàä 2.3.1. Îá÷èñëèòè âèçíà÷íèê<br />
1 2 3 4<br />
0 0 2 5<br />
∆=<br />
2 1 3 1 .<br />
1 2 1 7<br />
Ðîçâ’ÿçàííÿ. Îñê³ëüêè âèçíà÷íèê ìຠäâà íóë³ ó äðóãîìó<br />
ðÿäêó, òî ðîçêëàäåìî éîãî ïî åëåìåíòàõ öüîãî æ ðÿäêà<br />
124 123<br />
2+ 3 2+<br />
4<br />
∆= 0+ 0 + ( −1) ⋅2⋅ 211 + ( −1) ⋅5⋅213<br />
.<br />
127 1 21<br />
Ðîçêðèâàþ÷è âèçíà÷íèêè òðåòüîãî ïîðÿäêó çà áóäü-ÿêèì<br />
ïðàâèëîì, îòðèìàºìî ∆ =2⋅9+5⋅6=48.<br />
2.3.2. Âëàñòèâîñò³ âèçíà÷íèê³â<br />
Íàâåäåìî îñíîâí³ âëàñòèâîñò³ âèçíà÷íèê³â.<br />
1. Âåëè÷èíà âèçíà÷íèêà íå çì³íèòüñÿ, ÿêùî âñ³ éîãî<br />
ðÿäêè çàì³íèòè ñòîâïöÿìè ç òèì æå íîìåðîì, òîáòî<br />
T<br />
A = A .<br />
2. ßêùî ïîì³íÿòè ì³ñöÿìè äâà ñòîâïöÿ (ðÿäêè) âèçíà÷íèêà,<br />
òî éîãî çíàê çì³íèòüñÿ íà ïðîòèëåæíèé. Íàïðèêëàä,<br />
21 12 73<br />
∆ = = 6− 7= − 1; = 7− 6= 1; = 7− 6 1<br />
73 37 21<br />
= .<br />
3. Âèçíà÷íèê, ÿêèé ìຠäâà îäíàêîâèõ ñòîâïöÿ (ðÿäêè),<br />
äîð³âíþº íóëþ.<br />
ijéñíî, ÿêùî ó âèçíà÷íèêà äâà ñòîâïöÿ îäíàêîâ³, òî,<br />
ÿêùî ïîì³íÿºìî ¿õ ì³ñöÿìè, çíàê âèçíà÷íèêà çà âëàñòèâ³ñòþ<br />
2 ïîâèíåí çì³íèòèñÿ íà ïðîòèëåæíèé, à ñàì âèçíà÷íèê<br />
íå çì³íèòüñÿ, òîáòî ∆ =–∆ ⇒ 2∆ =0 ³ ∆ =0.<br />
4. ßêùî âñ³ åëåìåíòè áóäü-ÿêîãî ñòîâïöÿ (ðÿäêà) ì³ñòÿòü<br />
çàãàëüíèé ìíîæíèê, òî éîãî ìîæíà âèíåñòè çà çíàê âèçíà÷íèêà.<br />
Íàïðèêëàä,<br />
a11 ma12 a11 a12<br />
2 5<br />
= m ; = 20− 5= 15,<br />
a ma a a 110<br />
21 22 21 22<br />
2 5 2 5⋅1 2 1<br />
= = 5⋅ = 5(4− 1) = 15.<br />
110 1 5⋅<br />
2 12<br />
5. Âèçíà÷íèê, ó ÿêîãî åëåìåíòè äâîõ ñòîâïö³â (ðÿäê³â)<br />
â³äïîâ³äíî ïðîïîðö³éí³, äîð³âíþº íóëþ. ijéñíî, íåõàé<br />
Òîä³<br />
a a a ma<br />
11 12 11 11<br />
∆= =<br />
a21 a22 a21 ma<br />
.<br />
21<br />
a11 ma11 a11 a11<br />
= m = m⋅ 0 0<br />
a ma a a<br />
= .<br />
21 21 21 21<br />
6. ßêùî êîæíèé åëåìåíò áóäü-ÿêîãî ñòîâïöÿ (ðÿäêà) º<br />
ñóìîþ äâîõ äîäàíê³â, òî âèçíà÷íèê äîð³âíþº ñóì³ äâîõ âèçíà÷íèê³â,<br />
ó ÿêèõ ñòîâïöÿìè (ðÿäêàìè) º â³äïîâ³äí³ äîäàíêè,<br />
à ³íø³ åëåìåíòè çá³ãàþòüñÿ ç³ ñòîâïöÿìè (ðÿäêàìè) çàäàíîãî<br />
âèçíà÷íèêà:<br />
a + a a a a a a 2 + ( −1) ⋅4 5 2 5 −4 5<br />
'<br />
'<br />
11 11 12 11 12 11 12<br />
= + ;<br />
= +<br />
' '<br />
21<br />
+ a<br />
21 22 21<br />
a22<br />
3+<br />
2 7 3 7 2 7<br />
.<br />
21 22<br />
a a a a a<br />
Îñòàííÿ ð³âí³ñòü ä³éñíî ñïðàâåäëèâà:<br />
2 + (1)4 − ⋅ 5 25 −45<br />
=− 39, + = 14 −15 −28 − 10 =−39<br />
.<br />
3+<br />
2 7 3 7 2 7<br />
7. Âèçíà÷íèê íå çì³íèòüñÿ, ÿêùî äî åëåìåíò³â áóäü-ÿêîãî<br />
éîãî ñòîâïöÿ (ðÿäêà) äîäàòè â³äïîâ³äí³ åëåìåíòè äðóãîãî<br />
ñòîâïöÿ (ðÿäêà), ïîìíîæåí³ íà òå ñàìå ÷èñëî.<br />
Öÿ âëàñòèâ³ñòü âèïëèâຠç âëàñòèâîñòåé 3,4,6. ijéñíî,<br />
íåõàé<br />
a11 a12 a11 + ma12 a12<br />
∆= i ∆<br />
1<br />
=<br />
a a a + ma a<br />
.<br />
21 22 21 22 22<br />
54 55
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