Chiziqli algebraik tenglamalar sistemasini yechishning matritsa, Gauss va Gauss-Jordan usullari

X = A	B = −				2	11 −5			0		= −				10	+0−10		= −				0	^=	0
	12		12		12
									4	4 −4							20	+0−8				12		−1
							2
Demak, ^x1 ^= ^1, ^x2 ^= ^0 , ^x 3 ^=^−^1 yoki ^X ^= (^1;0;^−^1)^t																.

Agar sistema matritsasining rangi tenglama noma’lumlari sonidan kichik bo’lsa ham uning yechimini teskari matritsa usulida topish mumkin. Buni quyidagi misolda ko’rib chiqamiz.


2-misol. Ushbu chiziqli tenglamalar sistemasini teskari matritsa usulida yeching:


x₁ − 2x₂ + 3x₃ − 5x₄ = 2, 2x₁ + x₂ + 4x₃ + x₄ = −3, 3x₁ − 3x₂ + 8x₃ − 2x₄ = −1, 2x₁ − 2x₂ + 5x₃ − 12x₄ = 4
Yechish. Tenglamalar sistemasi matritsasi A va kengaytirilgan matritsasi ^{(A B)}

	1 −2			3 −5						1		−2	3	−5		2
		2141					, (A		B) =		2	1 4		1		−3
A =
	3		−3	8	−2						3	−3	8	−2		−1
		2	−2	5	−12						2	−2	5	−12		4

larning rangini topib

1		−2	3	−5		2		1		−23−5				2
	2	1 4		1		−3			0	5	−2 11			−7
	3	−3	8	−2		−1			0	3	−1	13		−7
	2	−2	5	−12		4			0	2	−1	−2		0

1		−2 3		−5		2	1			−2 3		−5
	0	5	−2	11		−7			0	5	−2 11
	0	0	1	32		−14			0	0	1	32
	0	0	−1	−32		14			0	0	0	0

2

−7

−14

WWW.OPENSCIENCE.UZ 314

"SCIENCE AND EDUCATION" SCIENTIFIC JOURNAL AUGUST 2021 / VOLUME 2 ISSUE 8
^{r (A ) = r (A B) = 3} ekanligini koʻramiz. Uning minori

=
1 −2 3

2. 
   1 4 =8−18−24−9+32+12=1
   

2. 
   −3 8
   

noldan farqli. Shuning uchun toʻrtinchi tenglamani tashlab yuboramiz, qolgan tenglamalarda ^x4 qatnashgan hadlarni oʻng tomonga oʻtkazamiz.

x₁ − 2x₂ + 3x₃ = 2 + 5x₄ , 2x₁ + x₂ + 4x₃ = −3 − x₄ , 3x₁ − 3x₂ + 8x₃ = −1 + 2x₄.

	Bu sistemani teskari matritsa usuli bilan yechamiz. Avval asosiy matritsa
teskarisini Gauss - Jordan usulida topamiz:
1		−2 3			1	0	0				1			−2 3				1 0					0						1				7 0				−803
														5 −2				−2 1															−1 0				41−2
	2	1 4			0	1	0					0											0								0
	3	−3 8			0	0	1					0		3 −1				−3 0					1								0		−3 1				30−1
						1		0			0		20 7 −11														20 7 −11
								1			0		−4−1 2									−1			=				−4				−1 2
						0															^,	A
								0			1		−9−3 5																−9				−3 5
						^0																															.
Tenglamalar sistemasining umumiy yechimni topish uchun