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Òàêèì ÷èíîì, âàðò³ñòü ìàòåð³àë³â ³ ðîáî÷î¿ ñèëè äëÿ<br />
ïîáóäîâè øêîëè ñòàíîâèòü 4090 ãðí., äëÿ ïîáóäîâè äèòÿ÷îãî<br />
ñàäêà — 2800 ãðí., äëÿ ïîáóäîâè æèòëîâîãî áóäèíêó<br />
— 3210 ãðí.<br />
3. Ùîá â³äïîâ³ñòè íà òðåòº ïèòàííÿ, ñêëàäåìî äîáóòîê<br />
ìàòðèöü Â, À ³ Ð.  ðåçóëüòàò³ îòðèìàºìî, ùî ÂÀÐ = 55160.<br />
Îòæå, âàðò³ñòü óñ³õ ñïîðóä ñòàíîâèòü 55160 ãðí. (ïåâíà<br />
ð³÷, óìîâíî).<br />
×èòà÷åâ³ íàïîëåãëèâî ðåêîìåíäóºìî: 1) ïåðåâ³ðèòè âñ³<br />
ïðîì³æí³ âèêëàäêè; 2) ïîÿñíèòè, ÷îìó äîáóòîê ÑÀ íå ìàº<br />
ñìèñëó; 3) ïîêàçàòè âëàñòèâ³ñòü àñîö³àòèâíîñò³ ìàòðèöü Â,<br />
À ³ Ð.<br />
Íà çàê³í÷åííÿ â³äì³òèìî îñíîâí³ âëàñòèâîñò³ ìíîæåííÿ<br />
ìàòðèöü.<br />
1. À⋅0 =0.<br />
2. À⋅Å = E⋅À=A (A — êâàäðàòíà ìàòðèöÿ).<br />
3. Ó çàãàëüíîìó âèïàäêó À⋅Â ≠ Â⋅À (À⋅Â ³ Â⋅À ìàþòü<br />
ñåíñ). ßêùî À⋅Â = Â⋅À, òî ìàòðèö³ íàçèâàþòüñÿ ïåðåñòàâíèìè<br />
(êîìóòàòèâíèìè).<br />
4. Íåõàé À, Â, Ñ — ìàòðèö³, ÿê³ ìîæíà äîäàâàòè ³ ïåðåìíîæóâàòè,<br />
³ α — ÷èñëî. Òîä³ ìàþòü ì³ñöå ð³âíîñò³<br />
α⋅(À⋅Â) =(α⋅À)⋅Â = À⋅(α⋅Â), À⋅Â⋅C =(À⋅Â)⋅C = À⋅(Â⋅C),<br />
À⋅(Â + C) =À⋅Â + À⋅C.<br />
5. (À⋅Â) Ò = Â Ò ⋅À Ò .<br />
Äîâåäåìî âëàñòèâ³ñòü 2, ïðè÷îìó íå îáìåæóþ÷è çàãàëüí³ñòü<br />
ïðè n =2.<br />
Íåõàé<br />
Òîä³<br />
A<br />
⎛a<br />
a ⎞ ⎛10⎞<br />
⎜ ⎟ ⎜ ⎟.<br />
⎝ ⎠ ⎝01⎠<br />
11 12<br />
= , E =<br />
a21 a22<br />
⎛a ⋅ 1+ a ⋅0 a ⋅ 0+ a ⋅1⎞ ⎛a a ⎞<br />
A⋅ E = =<br />
11 12 11 12 11 12<br />
⎜ ⎟ ⎜ ⎟<br />
⎝a21 ⋅ 1+ a22 ⋅0 a21 ⋅ 0+ a22 ⋅1⎠ ⎝a21 a22<br />
⎠ ,<br />
⎛1⋅ a + 0⋅a 0⋅ a + 1⋅a ⎞ ⎛a a ⎞<br />
E⋅ A= =<br />
11 12 11 12 11 12<br />
⎜ ⎟ ⎜ ⎟<br />
⎝1⋅ a21 + 0⋅a22 0⋅ a21 + 1⋅a22 ⎠ ⎝a21 a22<br />
⎠ ,<br />
çâ³äêè âèïëèâàº, ùî À⋅Å = E⋅À = A. Îòæå, âëàñòèâ³ñòü 2 ïðè<br />
n = 2 äîâåäåíî.<br />
²íø³ âëàñòèâîñò³ ïðè n = 2 ðåêîìåíäóºìî ÷èòà÷åâ³ äîâåñòè<br />
ñàìîñò³éíî.<br />
2.3. ÂÈÇÍÀ×ÍÈÊ, ̲ÍÎÐÈ É ÀËÃÅÁÐÀ¯×Ͳ<br />
ÄÎÏÎÂÍÅÍÍß<br />
2.3.1. Îçíà÷åííÿ<br />
Ðîçãëÿíåìî êâàäðàòíó ìàòðèöþ<br />
A<br />
⎛a a ... a<br />
⎜<br />
⎜<br />
a a ... a<br />
... ... ... ...<br />
⎜<br />
⎝a a ... a<br />
11 12 1n<br />
21 22 2n<br />
= ⎜ ⎟<br />
n1 n2<br />
nn<br />
⎞<br />
⎟<br />
⎟<br />
. (2.3.1)<br />
⎟<br />
⎠<br />
¯é ìîæíà ïîñòàâèòè ó â³äïîâ³äí³ñòü âèçíà÷åíå ÷èñëî, ÿêå<br />
íàçèâàºòüñÿ äåòåðì³íàíòîì, àáî âèçíà÷íèêîì ìàòðèö³.<br />
Ïîçíà÷àºòüñÿ âîíî ñèìâîëàìè<br />
a11 a12 ... a1n<br />
=∆<br />
a a ... a<br />
=<br />
... ... ... ...<br />
=<br />
a a ... a<br />
21 22 2n<br />
det A<br />
A<br />
A .<br />
n1 n2<br />
nn<br />
(2.3.2)<br />
Ïîíÿòòÿ âèçíà÷íèêà ââîäèòüñÿ ò³ëüêè äëÿ êâàäðàòíèõ<br />
ìàòðèöü. Ââîäèòüñÿ òàêîæ ïîíÿòòÿ ïîðÿäêó âèçíà÷íèêà,<br />
ÿêèé äîð³âíþº ïîðÿäêó ìàòðèö³.<br />
̳íîðîì äåÿêîãî åëåìåíòà êâàäðàòíî¿ ìàòðèö³ íàçèâà-<br />
ºòüñÿ âèçíà÷íèê íîâî¿ ìàòðèö³, ÿêó îòðèìóºìî ³ç äàíî¿ ìàòðèö³<br />
âèêðåñëþâàííÿì ðÿäêà ³ ñòîâïöÿ, ùî ïåðåòèíàþòüñÿ íà<br />
öüîìó åëåìåíò³, íàïðèêëàä ì³íîð a ij -ãî åëåìåíòà ìàòðèö³ À ïîçíà÷àºòüñÿ<br />
òàê:<br />
50 51
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