Guruh: 5es 22 kem bajardi: Usmanov Jasur Abdujabbar oʻgʻli Qabul qildi: Toshkent -2023 Reja: Matritsalar

^3^

1. 
   2 4_1_  2314 (10);
   

_

2. 
   _21_3 4__21_33 _21_44_ _63 _84_;
   

_
1 2 5 6 15  27 16  28 19 22

3. 
   3 47 8 35  47 36  48 43 50; 
   

 2 1   4. 03 4231 02 43   
 23 1(1) 2 2 10    33  4(1)  3 2  40 ^03  2(1) 0 2  20 	2 4 13   5    3 4  43  13 0 4  23 ^ ^  2

4 11

 6 0 .
0 6 

Agar A matritsaning satrlarini A₁,A₂,...,A_m bilan va B matritsaning ustularini B₁,B₂,...,B_n bilan belgilansa, u holda matritsalarni ko‘paytirish qoidasini quyidagi ko‘rinishda yozish mumkin:

 A1    C  AB   A...2 B1    Am 

B2

...

 A1B1  Bn  A...2B1   AmB1 

A1B2 A2B2 ... AmB2

... ... ... ...

A1Bn   A2Bn . ... ^ AmBn 

Matritsalarni ko‘paytirishda A⁴ yozuv ikkita bir xil matritsani ko‘paytmasini bildiradi: A²  A A. Shu kabi A³  A A A... Aⁿ  _A_A_...__A.
n marta
² 1 1
1.5-misol. f (x)2x  x 5 va A  _{0 2}_ bo‘lsin. f (A)ni toping.
_
Yechish. Matritsa ko‘rinishdagi f (A) funksiyaga o‘tishda  sonli qo‘shiluvchi I ko‘paytma bilan almashtiriladi, bu yerda I - birlik matritsa.
2  2 1 3 5 0 6 1
0 4 _ _0 4 _  _0 5_ _0 5_. _
Umuman olganda matritsalarni ko‘paytirish nokommutativ, ya’ni ABBA. Masalan, 1n o‘lchamli A matritsaning n1 o‘lchamli B matritsaga AB ko‘paytmasi sondan, ya’ni 11 o‘lchamli matritsadan iborat bo‘lsa, BA ko‘paytmasi n- tartibli kvadrat matritsa bo‘ladi.
Bir xil tartibli A va B kvadrat matritsalar uchun ABBA bo‘lsa, A va B matritsalarga kommutativ matritsalar, ABBA ayirmaga kommutator deyiladi. 1.6-misol. A  ^_0^{2 1}3^_ va B  _^34 ⁵1^__ matritsalarning kommutatorini toping.
_

Yechish.
AB 02 	1 3 34 	5 2314 251(1) 10 9  1 03 34 05 3(1) 12 3,
BA34 	5 2 10	1  32  50 31 53  6 18 3 42  (1)0 41 (1)3 8 1 , 

AB  BA_1012 _93_ _86 181 _ __44 _94_.
_
Matritsalarni ko‘paytirish amali ushbu xossalarga bo‘ysunadi ⁵.
1^o. A matritsa mn o‘lchamli va B,C matritsalar n p o‘lchamli bo‘lsa, A(B C)  AB AC bo‘ladi;
2^o. A matritsa mn o‘lchamli va B,C matritsalar n p o‘lchamli bo‘lsa, A(B C)  AB AC bo‘ladi;
3^o. A,B,C matritsalar mos ravishda mn,n p, pq o‘lchamli bo‘lsa,
A(BC)  (AB)C bo‘ladi;
4^o. (4) A,B,I,O moslashtirilgan matritsalar va , skalyar sonlar bo‘lsa, u holda:
1) (A)(B)  ()(AB); 2) A(B) (A)B (AB);
3) AI  IA A; 4) AOOAO; 5) (AB)^T  B^T A^T .
5^o. A,I,O n- tartibli kvadrat matritsalar va p,q manfiy bo‘lmagan butun
sonlar bo‘lsa, u holda:
1) A^pA^q A^pq; 2) (A^p )^q  (A) ^pq; 3) A¹  A; 4) A⁰  I.
Isboti. Xossalardan ayrimlarini ta’riflar yordamida isbotlaymiz va ayrimlarining to‘g‘riligiga misollarni yechish orqali ishonch hosil qilamiz.
2^o-xossani qaraylik. A (a_ij ) matritsa mn o‘lchamli va B  (b_ij ), C  (c_ij ) matritsalar n p o‘lchamli bo‘lsin. U holda 2 va 3-ta’riflarga ko‘ra istalgan i, j da birinchidan B  C  (b_ij  c_ij ) yoki
n n n n
A(B  C)  aik (bkj  ckj )  (aikbkj  aikckj )  aikbkj  aikckj
k1 k1 k1 k1
va ikkinchidan
n n
aikbkj  aikckj  AC  BC
k1 k1
bo‘ladi. Oxirgi ikkita tenglikdan A(B C)  AB AC bo‘lishi kelib chiqadi ⁶.
6^o- xossani qaraylik. A (a_ij ) va B  (b_ij ) bo‘lsin. Bundan A^T  (a_ij) va
B^T  (b_ij) bo‘ladi, bu yerda a_ij  a _ji , b_ij b_ji. U holda 3-ta’rifga ko‘ra istalgan i, j da birinchidan
n n
AB  a_ikb_kj yoki (AB)^T  a _jkb_ki
k1 k1 va ikkinchidan
n n n
a jkbki  bkia jk  bikaik  BT AT
k1 k1 k1
bo‘ladi. Bundan (AB)^T  B^TA^T bo‘lishi kelib chiqadi ⁷.
3^o-xossani to‘g‘riligiga misol yechish orqali ishonch hosil qilamiz.
3 1  2 4 1
A (1 2), B  __{0 4}_, C  _ _{5 0 2}_ bo‘lsin.
U holda

3 1  2 4 1 11 12 BC 0 4 5 0 2  20 0	1 , 8
11 12 1 A(BC)  1 2 20 0 8  29 12 3 1 AB  1 20 4  3 7, 	17,
 2 4 1 (AB)C  3 7 5 0 2  29 12 	17.

1. Mashqlar

Demak, A(BC)  (AB)C.
T

1.1. A kvadrat matritsa bo‘lsin. A  A simmetrik matritsa bo‘lishini ko‘rsating.
3 1 0 0
       
1.2. A4 matritsani X 0, Y 1 va Z 0 matritsalarning chiziqli
^5^_ 0 0 1
kombinatsiyasi ko‘rinishida ifodalang.
2 1 10
1.3. a₃_b_{ 1}_{ 5} _ bo‘lsa, a va b ni toping.
1.4. Matritsa 30 ta elementga ega bo‘lsa, u qanday tartiblarda berilishi mumkin?
1.5-1.1.8 masqlarda A,B matritsalar va , sonlar berilgan. AB matritsani toping:
1 1 1  2 3 1
1.5. A2 _3 0_, B_1 0 2_, 1,  2.
_
 0 3 1 2
   
1.6. A2 1, B 3 1,  2, 3.
 1 4_ _ 2 5_
 2 1 0  3 1 1
   
1.7. A1 3  2, B  0 1 0_, 3, 2.
 2 3 1   4 3 2

 2 1 2
 
1.8. A 5 3 3_, BE, 1, .
1 0 2
1.9. A va B moslashtirilgan matritsalar bo‘lsin. Quyidagilarni ko‘rsating:

1. 
   agar A matritsa satr matritsa bo‘lsa, u holda AB satr matritsa bo‘ladi;
   
2. 
   agar B matritsa ustun matritsa bo‘lsa, u holda AB ustun matritsa bo‘ladi.
   

1 2 x 0 x 0

10. 
    ^_{3 3}_^_{0 y}_  ^__{9 0}^bo‘lsa, x va y ni toping. _
    
11. 
    Agar A matritsa 33 o‘lchamli va ^{C esa} 55 o‘lchamli bo‘lsa, ABC ko‘paytma ma’noga ega bo‘lishi uchun B matritsa qanday o‘lchamda bo‘lishi kerak?
    

1 0

10. 
    A  _{1 0}_ matritsa berilgan. AB ko‘paytmani nol matritsaga aylantiruvchi B matritsani
    

_
toping.
1.13-1.1.16 mashqlarda A va B matritsalar berilgan. AB matritsani toping:
 2 0 1 1 2

10. 
    A_1 3_, B_3 2 0_.
    

2 1
  4 2

10. 
    A0 1, B_2 3_.
    

3 2_
1 1 4 1 3
   

10. 
    A3 0 1,B 0 1.
    

2 1 0  2 1
 1 1 2 4 0 2
   

10. 
    A2 0 3,B2 1 0 .
    

 1 1 0 0 1 3
2  2  1 4

10. 
    A  _₂ 3_, B  _{ 2 5}, C  B 3I bo^‘lsa, (AB)C matritsani toping.
    

3 1 4 5 1 4

10. 
    A  _₂ 4_, B  _{2 6}_, C  _ _{5 3}_ bo^‘lsa, A(BC)matritsani toping.
    

 4 2 3  0 1
    T 2

10. 
    A2 1 6,B 3 2 matritsalar berilgan. AB, B B, A matritsalarni
    

 1 2 2 2 0
toping.
1 2 ²

10. 
    A_{0 3}_ va f (x)  3x 5x 4 bo‘lsin. f (A) ni toping.
    

_
1 0 ²⁰

10. 
    A  _{1 1}_ bo‘lsa, A ni toping.
    

_

10. 
    Agar A²  I va A matritsa 22 o‘lchamli bo‘lsa, Ani toping.
    

Adabiyotlar

1. 
   Yo.U.Soatov. Oliy matematika 1-tom., T, “O’qituvchi” 1992
   
2. 
   Yo.U.Soatov. Oliy matematika 2-tom., T, “O’qituvchi” 1992
   
3. 
   Lay, David C. Linear algebra and is applications. Copyright. 2012, pp.162169.
   
4. 
   Kenneth L. Kuttler-Elementary Linear Algebra [Lecture notes] (2015). pp.
   

96-99.

5. 
   Sh.R.Xurramov ”Matematika” Toshkent- 2016.
   

1^ Lay, David C. Linear algebra and is applications. Copyright. 2012, pp. 92-112

2^ Lay, David C. Linear algebra and is applications. Copyright. 2012, pp. 92-112

3^ E.Kreyszig. Advancet engineering Matematics. Copyright. 2011, pp. 255-265

4^1 1 1 1 1 1 ^1 0^
f (A)  2A A  5I  20 2_ _0 2__0 2_  5_0 1_ 
_

5^ Lay, David C. Linear algebra and is applications. Copyright. 2012, pp. 92-112

6^ E.Kreyszig. Advancet engineering Matematics. Copyright. 2011, pp. 255-265

7^ E.Kreyszig. Advancet engineering Matematics. Copyright. 2011, pp. 255-265