4-Mavzu: Matritsalar va ularning ayrim хоssalari

Determinantlarni hisoblash.

1.Misol. Berilgan matritsaning determinantini hisoblang.
2 2 −1
^A=( 2 −4 1 ⁾
−2 −2 2

1. 
   usul. Determinantni ixtiyoriy satri yoki ustuni elementlari bo’yicha yoyib hisoblash teoremasidan foydalanamiz. Masalan, determinantni 3-ustun elementlari bo’yicha
   

yoyamiz

2	2	−1
2 −4 1
−2	−2	−2

:[ ]=−1⁽−1⁾¹⁺³
[ ^{2 −4}] + 1⁽−1⁾
−2 −2
2+3
[ ^{2 2} ] +
−2 −2

2(−1)³⁺³ [^{2 2}
2 −4
] = −⁽−4 − 8⁾ − ⁽−4 + 4⁾ + 2⁽−8 − 4⁾ = 12 − 0 − 24 = −12

2. 
   usul. Determinantni hisoblashning uchburchak qoidasidan foydalanamiz:
   

2 2 −1

^[ 2 −4 1
−2 −2 2
]=2· ⁽−4⁾ · 2 + 2 · 1 · ⁽−2⁾ + 2 · ⁽−2⁾ · ⁽−1⁾— 1 · ⁽−4⁾ ·

⁽−2⁾ −
2· 2 · 2 − ⁽−2⁾ · 1 · 2 = −16 − 4 + 4 − 8 + 8 + 4 = −12

3. 
   usul. Elementar almashtirishlar orqali determinantni biror satr yoki ustunni maksimal miqdorda nolli bo’lgan ko’rinishga keltiramiz va o’sha satr yoki ustun bo’yicha yoyamiz:
   

2 2 −1
^[ 2 −4 1
−2 −2 2
]=^{1 − 𝑠𝑎𝑡𝑟𝑔𝑎 3 − 𝑛𝑖 𝑞𝑜′𝑠ℎ𝑎𝑚𝑖𝑧^}=[
0 0 1
2 −4 1^]=
−2 −2 2

^{1 − 𝑠𝑎𝑡𝑟 𝑏𝑜^′𝑦𝑖𝑐ℎ𝑎 𝑦𝑜𝑦𝑜𝑚𝑖𝑧^}=1(−1)¹⁺³ [ ^{2 −4}]=-4-8=-12
−2 −2

4. 
   usul. Elementar almashtirishlar orqali uchburchak ko’rinishga keltiramz:
   

2 2 1

^[ 2 −4 1^{] {1 − 𝑠𝑎𝑡𝑟𝑛𝑖 − 1𝑔𝑎 𝑘𝑜′𝑝𝑎𝑦𝑡𝑖𝑟𝑖𝑏, 2 −}
−2 −2 2


2 2 −1

𝑠𝑎𝑡𝑟𝑔𝑎 𝑞𝑜^′𝑠ℎ𝑎𝑚𝑖𝑧𝑣𝑎 𝑛𝑎𝑡𝑖j𝑎𝑛𝑖 2 − 𝑠𝑎𝑡𝑟𝑔𝑎 𝑦𝑜𝑧𝑎𝑚𝑖𝑧^}=[
0 −6 2 ^]=
−2 −2 2

^{1 − 𝑠𝑎𝑡𝑟𝑔𝑎 3 − 𝑛𝑖 𝑞𝑜^′𝑠ℎ𝑎𝑚𝑖𝑧 𝑣𝑎 𝑛𝑎𝑡𝑖j𝑎𝑛𝑖 3 − 𝑠𝑎𝑡𝑟𝑔𝑎 𝑦𝑜𝑧𝑎𝑚𝑖𝑧^}
2 2 −1

^[ 0 −6 2
−2 −2 2
] =^{1 − 𝑠𝑎𝑡𝑟𝑔𝑎 3 − 𝑛𝑖 𝑞𝑜^′𝑠ℎ𝑎𝑚𝑖𝑧 𝑣𝑎 𝑛𝑎𝑡𝑖j𝑎𝑛𝑖 3 −

𝑠𝑎𝑡𝑟𝑔𝑎 𝑦𝑜𝑧𝑎𝑚𝑖𝑧^} =
2 2 −1

^[0 −6 2
0 0 1
] =^{𝑑𝑖𝑜𝑔𝑎𝑛𝑎𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑙𝑎𝑟𝑛𝑖 𝑘𝑜′𝑝𝑎𝑦𝑡𝑖𝑟𝑖𝑙𝑎𝑑𝑖^}=-12

5. 
   usul. A= ¹ (𝐴
   

· 𝐴
− 𝐴
· 𝐴
) formulada i=1 va j=2 bo’lsin. 𝐴
, 𝐴 ,

^𝐴𝑖𝑖jj
𝑖𝑖 jj
𝑖j
j𝑖
1122 11

𝐴₂₂, 𝐴₁₂, va 𝐴₂₁, - algebraik to’ldiruvchilarini hisoblaymiz:

𝐴₁₁₂₂ = 2 ; 𝐴₁₁ = [^{−4 1}]=-8+2=-6 𝐴₂₂ = [ ^{2 −1}] = 4 − 2 = 2

−2 2 −2 2
𝐴₁₂ = − [ ^{2 1}] = −⁽4 + 2⁾ = −6 𝐴₂₁ = − [ ^{2 −1}]=-(4-2)=-2
−2 2 −2 2
olingan natijalarni formulaga qo’yib, determinantning qiymatini topamiz:

A= ^1
𝐴𝑖𝑖jj
(𝐴_𝑖𝑖

• 
  𝐴_jj
  

− 𝐴_𝑖j

• 
  𝐴_j𝑖
  

)=¹(-6· 2 − (−6)(−2)) = −12
2

Berilgan ikkinchi tartibli determinantlarni hisoblang.

−7 6
10 −5
10 −5
_√𝑎 + √𝑏 _√𝑎 − √𝑏

1) [
_{5 −4}] 2) [ _{8 −8}] 3).[ _{9 −8}] 4) [√_{𝑎 −} √_𝑏 √_{𝑎 +} √_𝑏]

Berilgan uchinchi tartibli determinantni hisoblang.

	2	3	4		𝑎	1	𝑎		5	3	2		1	−1	4
1)	^[5	−2	1^]	2)	^[−1	𝑎	1^]	3)	^[−1	2	4^]	4)	^[3	−2	1	]
	1	2	3		𝑎	−1	𝑎		7	3	6		1	−1	−3

sin 𝑎	cos 𝑎	1		sin 𝑎	cos 𝑎	1	𝑐𝑡𝑔𝑏	𝑐𝑡𝑔𝑎	1
5) [^sin^𝑏	cos 𝑏	^1]	6)	[^sin^𝑏	cos 𝑏	^0]	7) [𝑠𝑖𝑛𝑎	𝑐𝑜𝑠𝑏	1]

sin 𝑦 cos 𝑦 1
sin 𝑦 cos 𝑦 0
𝑦 𝑦 1

Determinantning xossalaridan foydalanib, hisoblang.

1 2 3 4
−9 − 9 − 9 − 9
^{1) [} 4 3 2 1
1 0 1 0
− 1 − 1 − 1 − 1
_{] 2) [} −1 − 2 − 4 − 8
−1 − 3 − 9 − 27
−1 − 4 − 16 − 64

] 3) [

1 2 0 − 3
3 1 0 4 _]
1 5 − 1 7
−2 1 0 1

3 − 1 2 − 1 1
^⎡5 1 − 2 1 2^⎤
4) 9 − 1 1 3 4
3 − 1 2 − 1 1
^⎡ 5 1 − 2 1 2 ^⎤
5) 9 − 1 1 3 4

⎢ 3	0	6 − 1	3 ⎥	⎢3	− 1	2	− 1	1 ⎥
⎣ 5	2	3 − 2	1 ⎦	⎣ 5	2	3	− 2	1 ⎦

Nazorat misollari

1. 
   Kvadrat matritsalarga misollar.
   

−1 3 3
9 2 6 3

^{A=[1 −6] B=[}−1 2 4^{] С=[1 5 8 2]}

^{2 4} 8 7 1
3 2 1 4
1 0 3 0

Matritsalar mos ravishda 2-tartibli,3-tartibli,4-tartibli matritsalar deyiladi.

2. Uchburchak matritsalar.

2	−9	3
^D=^[0	−5	6^]
0	0	2

0	0
1	0^]
−1	3

8
^C=[−2
6

3. Diagonal matritsalar.

2	0	0	6			0	0
^A=^[0	−1	0^]	^B=^[0			9	0^]
0	0	9	0			0	5
4.Skalyar matritsalar. 5 0 0 −4 0 0
^A=^[0	5	0^]	B=[	0	−4			0	]	3-tartibli skalyar matritsalar.
0	0	5		0	0			−4
5.Birlik matritsalar.

_{1 0} 1 0 0
^A=[_{0 1}^{] B=[}0 1 0^]
0 0 1

6. Simmetrik matritsalar.

A=[
−8 3 −6
3 9 2
−6 2 −7
3 1 5
^{] B=[}1 9 4^]
5 4 7

7. Kososmmetrik matritsalar.

−2 −1 5 9 3 −8
^A=[ 1 5 6^{] D=[}−3 7 −4^]
−5 −6 4 8 4 1

1 −1		2		2		−1
^B=^[0	1	]	^C=^[0		3		4	]
1	0		0		0		4

Berilgan matritsalarning o’lchamlarini va turlarini aniqlang.

2 0 0 0

A=[ ^{2 4 1}]
−1 0 2
0 2 0 0 [ ]

D=
0 0 3 0
0 0 0 8

3				1	2
^Misol.^A=^[1				5	6^]	va 𝛼=4 uchun 𝛼 ·A					ni aniqlang.
1				6	4
3 1 2					12	4 8
𝛼 · 𝐴=4·	^[1 5		6^]	^=^[4		20	24^]
	1 6		4	4		24	16
	1		2	3		1	3		4
Misol	^.A=^[2		1	4^]^;		^B=^[5	7		8^]	matritsalar uchun 2A+B ni hisoblang.
	3		2	3		1	2		4
2 ^2A=^[4	4 6 2 8^]		2 ^2A+B=^[4			4 2	6 1 8^]^+^[5			3 7	4 2 + 1 4 + 3 6 + 4 8^]^=^[4 + 5 2 + 7 8 + 8^]^=
6 4		6	6			4	6		1	2 4		6 + 1	4 + 2	6 + 4
								3	7	10
								^[9	9	16^]
								7	6	10

Matritsalarni ko’paytiring.

A=[^{1 0 −1}] B=[
2 1 0
0 −1
1 0


] A· 𝐵 matritsani toping.

_[1 0 −1_{] · [}
2 1 0
−2 2
0 −1
1 0 ^]=
−2 2

_[1 · 0 + 0 · 1 + ⁽−1⁾ · ⁽−2⁾ 1 · ⁽−1⁾ + 0 · 0 + ⁽−1⁾ · 2_{] = [2 −3]}
2 · 0 + 1 · 0 + 0 · ⁽−2⁾ 2 · ⁽−1⁾ + 1 · 0 + 0 · 2 1 −2


1 0 3 1 −1
^A=[2 4 1^{] , B=[}3^{] , C=[} 2 ^{] matritsalar va 𝛼=2 soni uchun 𝐴𝑇B+𝛼C}

1 −4 2 2
matritsani aniqlang.
1 2 1
1
−1 −2

^𝐴𝑇=[0 4 −4^{] 𝛼 ·C=2[}
3 1 2
2 ^]=[ 4 ^{] ,}
1 2

1 2 1 1 1 · 1 + 2 · 3 + 1 · 2 9
^{𝐴𝑇𝐵=[}0 4 −4^{] · [}3^]=[0 · 1 + 4 · 3 − 4 · 2^]=[ 4 ^]
3 1 2 2 3 · 1 + 1 · 3 + 2 · 2 10
9 −2 7

^{𝐴𝑇B+𝛼C=[} 4 ^]+[
10
4 ^]=[ 8 ^]
2 12

1
^{Misol. A=[}4^{] va B=[}2 4 1^{] uchun AB va BA ko’paytmalarni aniqlang.}
3
1 1 · 2 1 · 4 1 · 1 2 4 1

^AB=^[4^]^·^[2	4	1^]^=^[4 · 2	4 · 4	4 · 1^]^=^[8	16	4^]
3		3 · 2	3 · 4	3 · 1 6	12	3

1
^BA=[2 4 1^{] · [}4^{]=2· 1 + 4 · 4 + 1 · 3 = 2 + 16 + 3 = 21}
3

Berilgan A, B matritsalarning o’lchamlarini aniqlang va 𝑎₁₃, 𝑎₂₃, 𝑎₂₁,

𝑏₂₂, 𝑏₁₂, 𝑏₁₃, 𝑏₂₃ elementlarini ko’rsating: A=‖ ^{2 4 1}‖, B=‖^{0 2 1}‖.
−1 0 2 1 1 2
Berilgan A, B matritsalarning o’lchamlarini aniqlang va 𝑎₁₃, 𝑎₁₂, 𝑏₃₁, , 𝑏₂₂, , 𝑏₃₂
4 −3
^{elementlarni ko’rsating: A=(1 2 3), B=‖}1 2 ^‖.
0 2
Matritsalar uchun ko’rsatilgan chiziqli amallarni bajariing.

1. 
   
   ‖
   A= ^{1 5}
   

2 −4
‖ B=
‖ ‖ 2A-B=?

3 2
4 1

2. 
   A=‖^{1 −1 −3}‖ B=‖ ^{0 3 2}‖ 3A-2B=?
   

2 1 5 −1 4 1

3. 
   A=‖^{1 −2}‖ B=‖ ^{3 2}‖ 2A-B=?
   

2 −5 −3 2

4. 
   
   1
   A=‖ ^{1 −2 −3}‖ B=
   

3 −3 2
‖ 3A-2B=?

−1 1 5
^‖ 4 −1

	3	5	7		1	2	4
5.	^A=^[2	−1	0^]	B=[	2	3	−2^]	A+B=?
	4	3	2		−1	0	1

6. 
   A=[^{3 5}] B=
   

4 1
2 3
[
1 −2
] 2A+5B=?

3 5 7

6. 
   ^A=[2 −1 1^{] B=[}
   

4 3 2
1 2 4
2 3 −1^{] A+B=?}
−1 1 1

Nazorat savollari

1. 
   Matritsani va uning asosiy tushunchalarini ta’riflang.
   
2. 
   Matritsaning asosiy turlarini sanab o’ting.
   
3. 
   Satr, ustun va kvadrat matritsalarni ta’riflang.
   
4. 
   Simmetrik va kososimmetrik matritsalarni ta’riflang
   

5. 
   Diagonal va birlik matritsalar qanday ko’rinishga ega?
   
6. 
   Uchburchak matritsalar qanday ko’rinishga ega.
   
7. 
   Qarama-qarshi va kengaytirilgan matritsalarni ta’riflang .
   
8. 
   Matritsani songa ko’paytirish (bo’lish) amali qanday bajariladi
   
9. 
   Matritsaning songa ko’paytirish (bo’lish) amalining xossalari.
   
10. 
    Matritsaning matrissaga qo’shish (ayirish) amali qanday bajariladi.
    
11. 
    Matritsani matritsaga qo’shish (ayirrish) amalining xossalarini keltiring.
    
12. 
    Matritsaning matritsaga ko’paytirish amalini ta’riflang.
    
13. 
    Matritsani matritsaga ko’paytirish amali xossalaring keltiring.
    
14. 
    Matritsani matritsaga ko’paytirish amalining zaruriy shartini keltiring.
    
15. 
    Matritsaning butun musbat darajaga ko’tarish qanday amalga oshiriladi?
    
16. 
    Matritsani butun musbat darajaga ko’tarish amalining xossalari.
    
17. 
    Transponirlangan matritsani ta’riflang.
    
18. 
    Transponirlash amali xossalarini keltiring.
    
19. 
    Determinant tushunchasi ta’riflang.
    
20. 
    Birinchi va ikkinchi tartibli determinantlarni ta’riflang.
    
21. 
    Uchinchi tartibli determinantni hisoblash qoidalarini bayon eting.
    
22. 
    Qanday matritsalar uchun determinant tushunchasi aniqlangan?
    
23. 
    Determinantning xossalarini ta’riflang.
    
24. 
    Determinant elementi minorini ta’riflang.
    
25. 
    Determinant elementi algebraik to’ldiruvchini ta’riflang.
    
26. 
    Elementar almashtirishlar qanday almashtirishlar?
    
27. 
    Kvadrat matritsa determinanti haqidagi teorema tastig’ini izohlang.
    
28. 
    Determinantlar hisoblash usullarini sanab o’ting.
    
29. 
    Xos va xosmas matritsaga ta’rif bering.
    
30. 
    Teskari matritsa tushunchasini ta’riflang.
    
31. 
    Teskari matritsa mavjudligi haqidagi teomera tastig’ini izohlang.
    
32. 
    Teskari matritsaning xossalarini bayon eting.
    
33. 
    Matritsa rangi tushunchasi.
    
34. 
    Matritsa rangini aniqlash usullari.
    
35. 
    Teskari matritsani aniqlash usullarini sanab o’ting va ta’riflang.
    
36. 
    O’lchami m n bo’lgan A matritsada nechanchi tartibli minorlari mavjud?