3-amaliy mashg‘ulot. Matritsa rangi. Teskari matritsa 1
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3-amaliy mashg‘ulot. Matritsa rangi. Teskari matritsa 3.1. Matritsa rangini ta’rifga asosan hisoblang: 2 1 3
2 4 4 2 5 1 7 .
2 1 1
8 2
matritsa 3 5
oʻlchamli, demak uning rangi 3 dan yuqori boʻlmaydi. Uchinchi tartibli minorlarni hisoblaymiz:
1 4 2 1 3 4 2 5
2 1 1
10 12 12 4 10 0; M
2 2 1 2 4 2 1 2 1 8 32 2 8 8 32 2 0; M
3 8 2 1 4 4 2 7
2 1 2
14 16 16 8 14 0; M
4 40 3 4 10 48 1 3 2
1 5 1 1 1 8 0; M
5 6 14 160 4 20 168 0;.. 3 2 4
5 1 7 1 8 2 . M
Barcha uchinchi tartibli minorlar nolga teng. Ikkinchi tartibli minorlarni hisoblaymiz:
1 1 1 1
5 6 1 0, 2 1 3 . 2 5
M M r A
Bu usulda noldan farqli minor topilgunga qadar hisoblashlar davom etadi. Shuning uchun 3 va undan kattaroq tartibli matritsa rangini hisoblash birmuncha qiyinchiliklarga olib keladi. 3.2. Matritsa rangini elementar almashtirishlar yordamida nollar yigʻib hisoblang: 25 31 17
43 75 94 53 132 75 94 54 134 25 32 20
48 A
Yechish: 25 31 17
43 25 31 17 43 25 31 17 43 75 94 53 132 0 1
3 0 1 2 3 . 75 94 54 134 0 1 3 5 0 0 1 2 25 32 20 48 0 1 3 5 0 0 0 0
Bu matritsaning rangi
25 31 17 0 1 2 0 0 1 matritsa rangiga teng. 25 31 17 0 1 2 0 0 1 25 0
25 31 17 0 1 2 0 0 1 3
Demak, berilgan matritsaning rangi ham 3 ga teng.
3. r A
3.3. Berilgan kvadrat matritsaning rangini toping. Xosmas matritsaning teskarisini toping: 1 2 2 5 ) ; ) ; 2 0 4 2
a A b B
3.4. 2 3 2 5 1 4 1 2 1 A matritsa uchun teskari 1
matritsani klassik usulda toping.
1, 2, 3; 1, 2, 3
ij A i j A matritsa elementlarining algebraik toʻldiruvchilari. 2 12 2
2 3 0 2 15 16 43 24 19 2 5 4 2 1 0 1 1
Demak, A xosmas matritsa, va 1 A teskari matritsa mavjud. Algebraik toʻldiruvchilarni hisoblaymiz:
11 1 4 2 1 1 8 7;
A
21 3 2 2 1 3 4 1; A
12 5 4 1 1 5 4 9; A
22 2 2 1 2 2 ; 1 4 A
13 10 1
11; 5 1 1 2
23 2 3 1 2 4 3
7; A
31 3 2 1 4 12 2 10;
A
32 2 2 5 8 10 4 2;
33 2 3 5 5 1 1 2 1 3 A
topilganlarni (2) formulaga qoʻyamiz va teskari 1 7 1 10 9 4 2 11 7 13 1/19 A
matritsani olamiz. Teskari matritsaning toʻgriligini tekshirish uchun quyidagi tenglikni tekshiramiz: 1 1
AA E A
(3) 2 3 2 7 1 10 5 1 4 9 1 / 1
4 2 1 2 1 11 7 13 9 14 27 22
2 12 14 20 6 26 35 9 44
5 4 28 50 2 52 7 18 11
1 8 7 10 4 13
19 0 0 1 0 0 0 19
0 0 1 0
0 0 19
0 0 1 1 / 19
1 / 19 E
Demak, 1 A toʻgʻri topilgan. 3.5. 1 2 1 1 1 3 4 3 2 A matritsa uchun 1
matritsani Gauss-Jordan usulida toping.
16 0
A
teskari matritsa mavjud. Berilgan matritsani birlik matritsa hisobida kengaytirib, elementar almashtirishlar bajaramiz, bu usulni to chap tomonda A matritsa oʻrnida birlik matritsa hosil boʻlguncha davom ettiramiz, oʻng tomonda hosil boʻlgan matritsa berilgan matritsaga nisbatan teskari matritsa boʻladi. 1 2 1 1 0 0 1 2 1 1 0 0
1 1 3 0 1 0 0 1 2 1 1 0 4 3 2 0 0 1 0 5 6 4 0 1
1 2 1 1 0 0
1 2 1 1 0 0 ~ 0 1 2 1 1 0 ~ 0 1 2 1 1 0 ~ 0 0 16 1 5 1
0 0 1 1/ 16
5 / 16 1/ 16
1 0 5 1 2 0 1 0 0 11 / 16 7 / 16 5 / 16
~ 0 1 0 14 / 16 6 / 16 2 / 16 ~ 0 1 0 14 / 16 6 / 16 2 / 16
0 0 1 1 / 16 5 / 16
1 / 16 0 0 1 1 / 16 5 / 16 1 / 16
1 11 7 5 14 6 2 1 5 1 1 / 16
A teskari matritsa toʻgʻri topilganini (3) formulaga qoʻyib tekshiramiz: 1 1 2 1 11 7 5 1 1 3 14 6 2 4 3 2 1 5 1 1 / 16 AA
11 28 1
7 12 5 5 4 1
11 14 3 7 6 15
5 2 3 44 42 2 28 1 / 16 18 10
20 6 2
= 16 0 0 1 0 0 0 16 0 0 1 0 0 0 16 0 0
1 1 / 16
demak, teskari matritsa toʻg’ri topilgan. 3.6. Berilgan kvadrat matritsalar uchun teskari matritsani ikki usulda toping: 1 1 1 3
) ; ) . 4 2 2 6 a b
3.7. Berilgan kvadrat matritsalar uchun teskari matritsani qulay usulda toping: 1 5 7
2 1 7
) 3 1 1 ; ) 5
3 2 ;
2 3 4 1 4 3 a b 3.8. Berilgan kvadrat matritsalarning rangini toping. Xosmas matritsaning teskarisini toping:
1 0 8 2 1 2 1 0 5 ) 5
9 0 ; ) 1 1 1 ; ) 4 2
0 4 3
2 3 2 2 1 3 a b c
2 3 4 0
1 2 1 0 1 5 7 0
1 1 3 1 ) ; ) 3 1 1 0
1 2 1 1 0 0 0 1
1 1 3 0 d e
Quyidagi matritsalar rangini minorlar ajratish usuli bilan hisoblang:
3 5 7 1 2 3 6 1 2 3
4 0 2 0 0
) 1 2 3 ; ) 2 3 1 6 ; ) 2 4 6 8 ;
) 1 0 0 4 1 3 5
3 1 2 6 3 6 9 12
0 0 3 0 a b c d
0 2 4 2 4 3 1 0 1 2 1 3 1 4 5 1 2 1 4 2
4 1 5 6 ) ; ) ; ) 3 1 7 0 1 1 3 1 1 3 4 7 0 5 10 4 7 4 4 5 2 1 1 0 2 3 0
j k
Misollarda matritsalar rangini elementar almashtirish usuli bilan hisoblang: 3.10. 1 2 1 3 4 3 4 2 6 8 ; 1 2 1 8 4
3.11. 1 7 5 8 9 2 3 21 15 24 27 6 ; 2 14 10 16 18 4 3.12. 1 2 3
4 2 4 6 8 ; 3 6 9 12
3.13. 1 0 2 0 0 0 1 0 2 0 ; 2 0 4 0 0
3.14. 4 3
5 2 3 8 6 7 4 2 ; 4 3 8 2
7 4 3
1 2 5 8 6 1 4
6
24 19 36 72 38 49 40 73 147 80 ; 73 59 98 219 118
47 36 71 141 72 3.16. 17 28 45 11 39 24 37 61 13 50 ; 25 7 32 18 11 31 12 19 43 55 42 13 29 55 68
47 67
201 155 26 98 23 294
6 ; 16 428 1 1284
52 3.18. 4 5 2 1 3 0 2 1 1
2 ; 4 7 3 3
1 8 12 5 3
4 3.19. 1 3 5 1 2 1 3 4 ; 5 1 1 7 7 7 9 1 3.20. 3 1 3 2 5 5 3 2 3 4 . 1 3 5 0 7 7 5 1 4 1 3.21. Berilgan kvadrat matritsalar uchun teskari matritsani ikki usulda toping:
2 1 1 1 ) ; ) 1 0 2 ; 2 3 1 2 tg a b ctg
1 1 1 2 5 7 3 4 5 ) 38 41 34 ; ) 6
3 4 ;
) 2 3 1 . 27 29 24 5 2 3 3 5 1 c d e
3.22. Berilgan kvadrat matritsalar uchun teskari matritsani qulay usulda toping:
1 0 2 3 2 2 ) 3 1 0 ; ) 1 3 1 ; 1 1 1
1 0 1 1
1 ) ; 0 0 1 1 0 0 0 1 1 2 4 5 3 4 a b c 2 3 2 1 2 1 0 0 0 0 0 1 1 0
0 0 0
0 1 0 1 0 0 0
) ; ) . 0 0
1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 n n n a a a a a a a a a d e a a a a
Foydalanishga tavsiya etiladigan adabiyotlar roʻyxati 1.
Mike Rosser. Basic mathematics for economists. London and New York. 1993, 2003.
2.
M.Harrison and P.Waldron. Mathematics for economics and finance. London and New York. 2011. 3.
M.Hoy, J.Livernois et. al. Mathematics for Economics. The MIT Press. London&Cambridge. 2011. 4.
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