Matritsalar algebrasi

M A S H Q L A R

56. A, B – bir xil tartibli diagonal matritsalar, α – son bo’lsin. isbot qilish lozimki, matritsalar ham diagonal matritsalardir va AB=BA.
57. bo’lsin. Isbot qilish lozimki:
1) BA matritsaning ustunlari V matritsaning ustunlarini sonlarga ko’paytirishdan hosil bo’ladi;
2) AB matritsaning satrlari B matritsaning satrlarini sonlarga ko’paytrishdan hosil bo’ladi.
58. A – diagonal matritsa, f(t) – ko’phad bo’lsin. u holda f(A) matritsa ham diagonal matritsa ekanligi isbotlang.
59. A matritsa diagonal matritsa bo’lib, uning hamma dioganal elementlari har xil bo’lsin. va AB=BA. U holda B matritsa ham diagonal matritsa ekanligi isbotlang.
60. A matritsa ixtiyoriy n-tartibli dioganal matritsa bilan o’rinalmashinuvchi bo’lsin. A matritsa n-tartibli dioganal matritsa ekanligini isbotlang.
61*. A matritsa hamma n-tartibli matritsaviy birliklar bilan o’rinalmashinuvchi bo’lsin. A matritsa skalyar matritsa ekanligini isbotlang.
62*. A matritsa n-tartibli har qanday matritsa bilan o’rinalmashinuvchi bo’lsin. A matritsa skalyar matritsa ekanligini isbotlang.
63. Berilgan matritsalarga qo’shma ermit matritsalar topilsin:
a) ; b) ; c) ; d) .
64. Ayniyatlarning to’g’riligini tekshiring:
a) ; b) ;
c) ; d) ; e) .
65. Berilgan ikkinchi tartibli matritsalar dioganal, skalyar, uchburchakli, simmetrik, kososimmetrik, ermit, kosoermit, unitar, ortogonal yoki o’rinalmashtirish matritsalari ekanligini aniqlang:
a) ; b) ; c) ; d) ;
e) ; f) ; g) ; h) .
66. Isbot qilingki:
a) kososimmetrik matritslarning hamma dioganal elementlari nolga teng;
b) ermit matritsasining dioganal elementlari haqiqiydir;
c) kosoermit matritsaning dioganal elementlari mavhum sondir;
67. Isbot qilish lozimki:
a) agar A ermit matritsasi bo’lsa, iA – kosoermit matritsasidir;
b) agar A kosoermit matritsasi bo’lsa, iA ermit matritsasidir.
68. a) ikkinchi tartibli ermit matritsalarning umumiy ko’rinishi topilsin;
b) ikkinchi tartibli kosoermit matritsalarning umumiy ko’rinishi topilsin;
c) hamma ikkinchi tartibli o’rinalmashtirishlarning matritsalarini ko’rsating.
69. Agar A diagonal matritsa bo’lib, uning hamma dioganal elementlari noldan farqli bo’lsa, teskari A^-1 matritsa mavjud va dioganal matritsa ekanligini isbotlang.
70. Agar A – maxsusmas simmetrik matritsa bo’lsa, A^-1 ham simmetrik matritsadir.
71. Agar A – maxsusmas kososimmetrik matritsa bo’lsa, A^-1 ham kososimmetrik matritsadir.