ЛЕКЦІЇ ² ВПРАВИ

3.3. Êîðèñòóþ÷èñü ïðàâèëîì Êðàìåðà, ðîçâ’ÿçàòè ñèñòåìó
ð³âíÿíü
ì 3x- 3y+ 2z
= 2,
ï
í4x- 5y+ 2z
= 1,
ïî
5x- 6y+ 4z
= 3.
³äïîâ³äü: (1, 1, 1).
3.4. Ðîçâ’ÿçàòè ìåòîäîì Ãàóññà ñèñòåìó ð³âíÿíü
ì x1+ 2x2 + 3x3
=
7,
ï
í2x1+ 3x2- x3
= 6,
ï ïî 3x1+ x2- 4x3
= 3.
³äïîâ³äü: (2, 1, 1).
3.5. Ðîçâ’ÿçàòè ìàòðè÷íèì ñïîñîáîì ñèñòåìè:
³äïîâ³äü: (2, 1, 1).
ì x- 2y+ z = 1,
ï
í2x+ y- z = 4,
ïî
x- y+ 2z
= 3.
ì 2x1+ x2 + 3x3 + 4x4
=
11,
ï7x1+ 3x2 + 6x3 + 8x4
= 24,
í
3x1+ 2x2 + 4x3 + 5x4
= 14,
ï
ïî x1 + x2 + 3x3 + 4x4
= 11.
³äïîâ³äü: (1, –1, 2, 1).
3.6. ×è ñóì³ñíà ñèñòåìà ð³âíÿíü? ßêùî ñóì³ñíà, òî ðîçâ’ÿçàòè
¿¿:
ì
ï3x1+ 2x2 + x3+ x4
= 1,
í
ï
ïî 3x1+ 2x2-x3- 2x4
= 2.
³äïîâ³äü: (ñ 1 , ñ 2 ,4−9ñ 1 −6ñ 2 ,6ñ 1 +4ñ 2 −3).
ì x1- x2 + x3- 2x4
= 1,
ï
íx1 - x2 + 2x3 - x4
= 2,
ïî
5x1- 5x2 + 8x3- 7x4
= 3.
³äïîâ³äü: íåñóì³ñíà.
3.7. Âèçíà÷èòè ë³í³éíî íåçàëåæí³ ðîçâ’ÿçàííÿ ñèñòåìè
ð³âíÿíü
ì 2x1+ 3x2 + x3+ 2x4- x5
=
0,
ï
í3x1+ x2- 8x3+ 3x4 + 2x5
= 0,
ï ïî x1+ 2x2 + 2x3+ x4 + 6x5
= 0.
Âêàç³âêè:
1. Çíàéòè ðàíã ñèñòåìè (â³í äîð³âíþº 3).
2. Âèðàçèòè õ 1 , õ 2 , õ 3 ÷åðåç õ 4 ³ õ 5 .
3. Ïîêëàñòè õ 4 = 1, õ 5 =0 ³ õ 4 = 0, õ 5 =1.
4. Çíàéòè â³äïîâ³äí³ ÷èñëîâ³ çíà÷åííÿ õ 1 , õ 2 , õ 3 ³ õ' 1 , õ' 2 ,
õ' 3 . Ñêëàñòè ë³í³éíó êîìá³íàö³þ õ = l 1 ó 1 + l 2 ó 2 , äå
ó 1 =(õ 1 , õ 2 , õ 3 , 1, 0), ó 2 =(x 1 , x 2, x 3 , 0, 1).
3.8. Ðîçâ’ÿçàòè òàê³ ñèñòåìè ð³âíÿíü ìàòðè÷íèì ìåòîäîì,
çà ïðàâèëîì Êðàìåðà ³ çà ìåòîäîì Ãàóññà:
³äïîâ³äü: (1, 2, 3).
³äïîâ³äü: (1, 1, 1).
ì Nx1- Nx2 + Nx3
= 2 N,
ï
íNx1+ Nx2- Nx3
= 0,
ïî
Nx1-Nx2- Nx3
=-4 N.
ì Nx1+ x2 + Nx3
= 2N
+ 1,
ï
í - x1 + Nx2 + x3
= N,
ïî
Nx1- x2 + Nx3
= 2N
-1.
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