ЛЕКЦІЇ ² ВПРАВИ

Òîä³
a
12 12
∆
1
=∆+ m =∆ a
22 a
.
22
8. Ñóìà äîáóòê³â åëåìåíò³â a ij äåÿêîãî ðÿäêà (ñòîâïöÿ)
âèçíà÷íèêà íà àëãåáðà¿÷í³ äîïîâíåííÿ åëåìåíò³â ³íøîãî
ðÿäêà (ñòîâïöÿ) äîð³âíþº íóëþ:
n
n
ik jk ki kj
k= 1 k=
1
∑ a A = ∑ a A = 0, i ≠ j, i, j = 1, n.
Öþ âëàñòèâ³ñòü ìè äîâåäåìî ó çàãàëüíîìó âèïàäêó. Ïîðÿä
³ç êâàäðàòíîþ ìàòðèöåþ (2.3.1) ðîçãëÿíåìî äîïîì³æíó
ìàòðèöþ A , ÿêà áóäóºòüñÿ ³ç ìàòðèö³ A çàì³íîþ j-ãî ðÿäêà
íà ³-é ðÿäîê (i≠j, i, j = 1, n ):
a
⎛a11 a12 ... a1
n ⎞
⎜
⎟
⎜
... ... ... ...
⎟
⎜a1 2
... ⎟
i
ai ain
⎜
⎟
A = ⎜... ... ... ... ⎟
⎜a1 2
... ⎟
i
ai ain
,
⎜
⎟
⎜... ... ... ... ⎟
⎜
1 2
...
⎟
⎝an an ann⎠
òîáòî ìàòðèöÿ A ìຠäâà îäíàêîâèõ ðÿäêè. ßñíî, ùî òàêó
âëàñòèâ³ñòü áóäå ìàòè ³ â³äïîâ³äíèé âèçíà÷íèê A . Çã³äíî
ç âëàñòèâ³ñòþ 3 ¿¿ âèçíà÷íèê äîð³âíþº íóëþ. Ç äðóãîãî
áîêó, îá÷èñëèìî âèçíà÷íèê A ñïîñîáîì ðîçêëàäàííÿ çà
åëåìåíòàìè j-ãî ðÿäêà, ó ðåçóëüòàò³ îòðèìàºìî:
n
A = ∑ aik
Ajk
= 0, i ≠ j, i, j = 1, n.
k=
1
Òàêèì ÷èíîì, ïåðøà ÷àñòèíà âëàñòèâîñò³ 8 äîâåäåíà.
Àíàëîã³÷íî äîâîäèòüñÿ äðóãà ÷àñòèíà âëàñòèâîñò³ 8.
Ïîäàìî áåç äîâåäåííÿ ùå îäíó âëàñòèâ³ñòü âèçíà÷íèêà
ìàòðèö³, ÿêà º äîáóòêîì äâîõ êâàäðàòíèõ ìàòðèöü.
9. Âèçíà÷íèê äîáóòêó äâîõ êâàäðàòíèõ ìàòðèöü äîð³âíþº
äîáóòêó ¿õ âèçíà÷íèê³â:
C = A ⋅ B , äå C = AB; A i B — ìàòðèö³ n-ãî ïîðÿäêó.
³äçíà÷èìî, ùî âèêîðèñòàííÿ âëàñòèâîñòåé âèçíà÷íèêà
ñïðîùóº éîãî îá÷èñëåííÿ.
Ïðèêëàä 2.3.2. Îá÷èñëèòè âèçíà÷íèê
123
∆= 456
.
789
Ðîçâ’ÿçàííÿ. Âèçíà÷íèê áóäåìî îá÷èñëþâàòè òðüîìà
ñïîñîáàìè.
1 0 (áàçóºòüñÿ íà ïðàâèë³ òðèêóòíèê³â). Âèêîðèñòîâóþ÷è
éîãî, îòðèìàºìî:
123
∆ = 456= 159 ⋅ ⋅ + 267 ⋅ ⋅ + 483 ⋅ ⋅ −753 ⋅ ⋅ −429 ⋅ ⋅ −861 ⋅ ⋅ = 0.
789
2 0 (áàçóºòüñÿ íà ðîçêëàäàíí³ âèçíà÷íèêà ïî åëåìåíòàõ
ðÿäê³â òà ñòîâïö³â). Çã³äíî ç ôîðìóëîþ 2.3.4 ìàºìî:
123
56 46 45
∆= = − ⋅ + − ⋅ ⋅ + − ⋅ =
89 79 78
789
1+ 1 1+ 2 1+
3
456 (1) (1) 2 (1) 3 0.
3 0 (áàçóºòüñÿ íà âëàñòèâîñòÿõ âèçíà÷íèêà). Çàñòîñîâóþ÷è
ïîñë³äîâíî ïðàâèëà 7,4 áóäåìî ìàòè:
123 1 2 3
−3 −6 1 2
∆= 456= 0−3 − 6 = = (3)(6) − ⋅ − = 0.
−6 −12 1 2
789 0−6−12
56 57