4-Mavzu: Matritsalar va ularning ayrim хоssalari

Determnant tushunchasi.

Deteriminant-berilgan sonli jadval elementlari asosida maxsus qoida bo’yicha hisoblanadigan son bo’lib,jadvalning tartibiga qarab,uni hisoblash qoidasi aniqlanadi.Matritsa va determinantlar uchun o’zaro farqli belgilashlardan foydalaniladi. Masalan, A matritsaning determinant |A| yoki detA orqali belgilanadi.Ixtiyoriy haqiqiy sonni birinchi tartibli tartibli determinant deb qarash mumkin.

1. 
   Tarif: Ikkinchi tartibli [^{𝑎 𝑏}] matritsaga mos keluvchi va a·d-b·c
   

𝑐 𝑑
munosabatlar bilan aniqlangan songa 2-tartibli determinant deyiladi va |^{𝑎 𝑏}|
𝑐 𝑑
kabi belgilanadi.
Determinant uchun satr ,ustun,element,tartib va diagonal tushunchalari kvadrat matritsalardagi kabi anqlangan.Tarifga ko’ra 2-tartibli determinantning qiymati asosiy dioganalda yotuvchi ikki element ko’paytmasidan yordamchi diagonalda yotuvchi ikki element ko’paytmasini ayirish natijasiga teng:
𝑎 𝑏
|_{𝑐 𝑑}| = 𝑎𝑑 − 𝑏𝑐
Misol. Quyidagi ikkinchi tartibli determinantni hisoblang.
|^{3 −4}| = 3 · 5 − ⁽−4⁾ · 2 = 15 + 8 = 23
2 5
^𝑎11 ^𝑎12 ^𝑎13

2. 
   ^{Tarif: Uchinchi tartibli (𝑎}21 ^𝑎22 ^𝑎23^{) matritsa elementlari asosida}
   

^𝑎31 ^𝑎32 ^𝑎33
𝑎₁₁ · 𝑎₂₂ · 𝑎₃₃+𝑎₁₂ · 𝑎₂₃ · 𝑎₃₁+𝑎₁₃ · 𝑎₂₁ · 𝑎₃₂-𝑎₁₃ · 𝑎₂₂ · 𝑎₃₁ −
^𝑎11 ^{· 𝑎}23 ^{· 𝑎}32 ^{− 𝑎}12 ^{· 𝑎}21 ^{· 𝑎}33

^𝑎11	^𝑎12	^𝑎13
^|^𝑎21	^𝑎22	^𝑎23^|
^𝑎31	^𝑎32	^𝑎33

qoida bilan hisoblangan son uchinchi tartibli determinant deyiladi va

kabi belgilanadi.

Hisoblashni soddalashtirish maqsadida dastlabki uchta qo’shiluvchi va keyingi uchta ayriluvchilar mos ravishda quydagi sxema bo’ycha hisoblanadi

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_{≪ +≫ (}· · ·_{) (}· · ·_{) (}· · ·₎

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_{≪ −≫ (}· · ·_{) (}· · ·_{) (}· · ·₎

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Bayon etilgan bu qoida uchinchi tartibli determinantni hisoblashning
uchburchak qoidasi deyiladi.
Misol. Berilgan uchinchi tartibli determinantning hisoblanishiga e’tibor bering.

2 1 −2
^{𝑎) |}5 0 3
4 −3 6
|=2· 0 · 6 + 1 · 3 · 4 + 5 · ⁽−2⁾ · ⁽−3⁾ − 4 · 0 · ⁽−2⁾ − 1 · 5 ·

6 − 2 · 3 · ⁽−3⁾ = 0 + 12 + 30 + 0 − 30 + 18 = 30

1 1 1
^{𝑏) |}2 −3 1
4 −1 −5

|=1· ⁽−3⁾ · ⁽−5⁾ + 1 · 1 · 4 + 2 · ⁽−1⁾ · 1 − 1 · ⁽−3⁾ · 4

−1 · 2 · ⁽−5⁾ − 1 · ⁽−1⁾ · 1 = 15 + 4 − 2 + 12 + 10 + 1 = 40

3. 
   Ta’rif: Ixtiyoriy kvadrat matritsa elementlari asosida ma’lum bir qoida asosida aniqlangan son bu matritsaning determinanti (aniqlovchisi)deb ataladi.
   

Biroq bu qoidalar uchinchidan yuqjri tartibli determinantlar uchun ishlamaydi.Shu sababli quyida determnantni hisoblashning universal usulini keltiramiz.
Birinchi tartibli kvadrat A=⁽𝑎₁₁⁾ matritsaning aniqlovchisi sifatida shu sonning o’zi olinadi, yani ^|𝐴^|=𝑎₁₁.

𝑎 𝑎
Ikkinchi tartibli kvadrat A=(^{𝑎11 𝑎12}) matritsaning aniqlovchisi sifatida
21 22
quydagi qoida asosida hisoblangan sonni qabul qilamiz, ya’ni:

^|𝐴^|=|^{𝑎11 𝑎12}|=𝑎
𝑀 -𝑎
𝑀 .

^𝑎21 ^𝑎22
11 11
12 12

Bunda 𝑀_𝑖j-berilgan A matritsada i-satr va j-ustunni o’chirishdan hosil bo’lgan
birinchi tartibli kvadrat matritsalarning determinantlari.
Masalan, 𝑀₁₁determinant A matritsada 1-satr va 1-ustunni o’chirishdan hosil bo’lgan .𝑀₁₂ determinant A matritsada 1-satir va 2-ustunni o’chirishdan hosil bo’lgan.

Masalan, A=(
1 0 −1
2 1 0
) matritsaning determinanti quydagiga teng:

−2 1 5
1 0 −1

^|𝐴^|=|
2 1 0
−2 1 5
|=1· 1 · 5 + 0 · 0 · ⁽−2⁾ + 2 · 1 · ⁽−1⁾ − (−2) · 1 · ⁽−1⁾ −

1 · 1 · 0 − 0 · 2 · 5 = 1


^𝑎11 ^𝑎12 ^{⋯ 𝑎}1𝑛

4. 
   
   ⋯ ⋯ ⋯ ⋯
   Ta’rif: n-tartibli kvadrat A=[ ^{𝑎21 𝑎22 ⋯ 𝑎2𝑛} ] matritsa determinanti
   

^𝑎𝑛1 ^𝑎𝑛2 ^{⋯ 𝑎}𝑛𝑚
deb quydagicha aniqlangan songa aytiladi:
^𝑎11 ^𝑎12 ^{⋯ 𝑎}1𝑛

^|𝐴^|=| ^{𝑎21 𝑎22 ⋯ 𝑎2𝑛} |=𝑎 𝑀
− 𝑎
𝑀 +. . . +(−1)^𝑛+1𝑎 𝑀
(2)

⋯ ⋯ ⋯ ⋯
^𝑎𝑛1 ^𝑎𝑛2 ^{⋯ 𝑎}𝑛𝑚
11 11
12 12
1𝑛
1𝑛

5. 
   Ta’rif: n-tartibli A matritsaning i-satr va j-ustunini o’chirishdan hosil bo’lgan (n-1)-tartibli determinantga A matritsa 𝑎_𝑖j elementining 𝑀_𝑖j minori deb ataladi.
   

Masalan,uchinchi tartibli A matritsa 𝑎₂₃ elementning 𝑀₂₃ minori A matritsadan 2-satr va 3-ustunni o’chirishdan hosil bo’lgan son.

6. 
   Ta’rif: n-tartibli A matritsaning 𝑎_𝑖j elementining 𝐴_𝑖j algebrik to’ldiruvchisi deb uning 𝑀_𝑖j minorini (−1)^𝑖+j ga ko’aytirilganiga aytiladi, ya’ni 𝐴_𝑖j=(−1)^𝑖+j𝑀_𝑖j
   

^𝑎11 ^𝑎12 ^{⋯ 𝑎}1𝑛

⋯ ⋯ ⋯ ⋯
Algebrik to’ldiruvchining ta’rifidan foydalanib | ^{𝑎21 𝑎22 ⋯ 𝑎2𝑛} |
^𝑎𝑛1 ^𝑎𝑛2 ^{⋯ 𝑎}𝑛𝑚
formulani quydagi ko’rinishda yozish mumkin:
^|𝐴^|=𝑎₁₁𝑀₁₁ − 𝑎₁₂𝑀₁₂+. . . +(−1)^𝑛+1𝑎_1𝑛𝑀_1𝑛 = 𝑎₁₁𝐴₁₁ + 𝑎₁₂𝐴₁₂+…+𝑎_1𝑛𝐴_1𝑛
1 2 1
^{Misol: Berilgan A=[}0 −2 3^{] matritsa determinantini hisoblang.}
3 1 1

Yechish: Determinantni matritsa birinchi satr elementlari bo’ycha yoyib hisoblaymiz.

1 2 1
−2 3 0 3
0 −2

^|0 −2 3^{|=1· |}
3 1 1
|-2· |
1 1 3 1
|+1· | |=(-2· 1 − 1 · 3)-2(0· 1 − 3 ·
3 1

3. 
   +(0· 1 + 3 · 2)=-5+18+6=19
   

Determinantning xossalari.

1. 
   xossa. Determinantning biror satr (ustuni) nollardan iborat bo’lsa, bu determinantning qiymati nolga teng.
   

^𝑎11 ^𝑎12 ^𝑎13

_[ 0 0 0
]=0

^𝑎31 ^𝑎32 ^𝑎33

2. 
   xossa. Determinantning qiymatini biror songa ko’paytirish uning biror satri (ustuni) elementlarini shu songa ko’paytirishga teng kuchli.
   

^{𝑎11 𝑎12 𝑎13 𝑎}11 ^𝑎12 ^𝑎13
^[𝑘𝑎21 ^𝑘𝑎22 ^𝑘𝑎23^{] = 𝑘 [𝑎21 𝑎22 𝑎23]
𝑎}31 ^𝑎32 ^𝑎33 ^{𝑎31 𝑎32 𝑎33}

3. 
   xossa. Matritsani transponirlash natijasida uning determinantining qiymati o’zgarmaydi , yani berilgan A matritsa uchun ^|𝐴^| = ^|𝐴^𝑇|.
   

^𝑎11 ^𝑎12 ^𝑎13 ^𝑎11 ^𝑎21 ^𝑎31
^[𝑎21 ^𝑎22 ^𝑎23^]=[𝑎12 ^𝑎22 ^𝑎32^]
𝑎31 ^𝑎32 ^𝑎33 ^𝑎13 ^𝑎23 ^𝑎33

4. 
   xossa. Determinantda ikkita satr yoki ikkita ustunning o’rni almashtrilsa,determinantning qiymati o’z ishorasini almashtiradi.
   

^𝑎11 ^𝑎12 ^𝑎13 ^𝑎21 ^𝑎22 ^𝑎23
^[𝑎21 ^𝑎22 ^𝑎23^{]=− [𝑎}11 ^𝑎12 ^𝑎13^]
𝑎31 ^𝑎32 ^𝑎33 ^𝑎31 ^𝑎32 ^𝑎33

5. 
   xossa.Ikkita bir xil satrga (ustunga) ega bo’lgan determinant nolga teng.
   

^𝑎11 ^𝑎12 ^𝑎13
^[𝑎11 ^𝑎12 ^𝑎13^{] = 0
𝑎}31 ^𝑎32 ^𝑎33

6. 
   xossa.Determinantning ikkita satr (ustuni) o’zaro proportsional bo’lsa,bu determinantning qiymati nolga teng.
   

^𝑎11 ^𝑎12 ^𝑎13
^[𝑘𝑎11 ^𝑘𝑎12 ^𝑘𝑎13^{] = 0
𝑎}31 ^𝑎32 ^𝑎33

7. 
   xossa. Agar determinantning biror satri (ustuni) to’laligicha ikki qo’shiluvchining yig’indisidan iborat bo’lsa , uning qiymatini ikkita determinantlar qiymatlari yig’indisi tarzida hisoblash mumkin.
   

^𝑎11 ^𝑎12 ^𝑎13
^𝑎11 ^𝑎21 ^𝑎31
^𝑎11 ^𝑎21 ^𝑎31

^[𝑎21 ^{+ 𝑏}1 ^𝑎22 ^{+ 𝑏}2 ^𝑎23 ^{+ 𝑏}3^{] = [𝑎12 𝑎22 𝑎32] + [ 𝑏}1 ^𝑏2 ^𝑏3 ^]

^𝑎31 ^𝑎32 ^𝑎33
^𝑎13 ^𝑎23 ^𝑎33
^𝑎13 ^𝑎23 ^𝑎33

8. 
   xossa. Agar determinantnng biror satri elementlarini bir xil songa ko’paytirib boshqa biror satrga qo’shilsa uning qiymati o’zgarmaydi.
   

^{𝑎11 𝑎12 𝑎13 𝑎}11 ^𝑎12 ^𝑎13
^[𝑘𝑎11 ^{+ 𝑎}21 ^𝑘𝑎12 ^{+ 𝑎}22 ^𝑘𝑎13 ^{+ 𝑎}₂₃^{] = [𝑎21 𝑎22 𝑎23]
𝑎}31 ^𝑎32 ^𝑎33 ^{𝑎31 𝑎32 𝑎33}

9. 
   xossa. Matritsa biror satr elementlarini boshqa biror satr elementlarining algebraik to’ldiruvchilariga ko’paytmalarining yig’indisi nolga teng.
   

^𝑎11^𝐴31 ^{+ 𝑎}12^𝐴32^+𝑎13^𝐴33^{=0; 𝑎}21^𝐴31^+𝑎22^𝐴32^+𝑎23^𝐴33⁼⁰

10. 
    xossa. Kvadrat matritsalar ko’paytmasining determinant har bir matritsa determinantlari ko’paytmasiga teng,ya’ni ^|𝐴 · 𝐵^| = ^|𝐴^| · ^|𝐵^|
    

Ixtiyoriy A va B kvadrat matritsalar uchun det(AB)=detA·detB
Sonli jadvallar(matritsa,determinant) elementlari ustidadagi elementar almashtirishlar deganda satr elementlarini biror songa ko’paytirib boshqa bir satrga qo’shish ,satrlar o’rnini almashtirish tushuniladi.