Abdishukurov shaxzod

TOSHKENT DAVLAT IQTISODIYOT UNIVERSITETI SAMARQAND FILIALI ST-121 GURUH TALABASI ABDISHUKUROV SHAXZODNING AMALIY MATEMATIKA FANIDAN MUSTAQIL TA`LIM PREZENTATSIYASI
Bajardi: Abdishukurov Shoxzod
Tekshirdi: Ismoilov G`olib
MAVZU: MATRITSANING XOSSALARI
REJA:

• MATRITSA TUSHUMCHASI
• MATRITSA USTIDA AMALLAR

Matritsalar va ular ustida amallar

𝑚x 𝑛 dona 𝑎𝑖 𝑗 (𝑖 = 1, 𝑚, 𝑗 = 1, 𝑛 ) elementlardan tuzilgan to’g’ri burchakli jadval matritsa deyiladi va

	𝑎11	𝑎12	𝑎13	. . .	𝑎1𝑛		𝑎11	𝑎12	𝑎13	. . .	𝑎1𝑛
𝐴 =	𝑎21 . . .	𝑎22 . . .	𝑎23 . . .	. . . . . .	𝑎2𝑛 . . .	yoki 𝐴 =	𝑎21 . . .	𝑎22 . . .	𝑎23 . . .	. . . . . .	𝑎2𝑛 . . .
	𝑎𝑛1	𝑎𝑛2	𝑎𝑛3	. . .	𝑎𝑛𝑛		𝑎𝑛1	𝑎𝑛2	𝑎𝑛3	. . .	𝑎𝑛𝑛

ko’rinishda yoziladi. Matritsaning elementlari ikkita indesklar bilan belgilanadi. Elementning birinchi 𝑖 indeksi satr nomini, ikkinchi 𝑗 indeks esa ustunning nomerini bildiradi. Matritsaning 𝑎𝑖 𝑗 elementi 𝑖 − satr va 𝑗 − ustun kesishgan joyda joylashgan.
Matritsalar odatda katta lotin harflari bilan belgilanadi:
𝐴, 𝐵, 𝐶, . . .

Matritsalar va ular ustida amallar

Agar matritsa 𝑚 ta satr va 𝑛 ta ustunga ega bo’lsa, u holda ta’rifga binoan, bu matritsa
𝑚× 𝑛 o’lchovga ega bo’ladi. Zaruriyat bo’lganida matritsani 𝐴𝑚×𝑛 ko’rinishda ham belgilaymiz. Agar matritsaning 𝑎𝑖 𝑗 elementlari sonlar bo’lsa, bunday matritsa sonli
matritsa deyiladi; agar matritsaning 𝑎𝑖 𝑗 elementlari funksiyalar bo’lsa, bunday matritsa
funksional matritsa deyiladi; 𝑎𝑖 𝑗 elementlar vektorlar bo’lganda esa, vektor matritsa deyiladi va hokazo.

Matritsalar va ular ustida amallar

Agar 𝐴 va 𝐵 matritsalarning mos 𝑎𝑖 𝑗 va 𝑏𝑖 𝑗 elementlari bir-biriga teng, ya`ni
𝑎𝑖 𝑗 = 𝑏𝑖 𝑗 bo’lsa, bunday 𝐴 va 𝐵 matritsalar teng matritsalar deyiladi. Faqat bir xil o’lchovli matritsalargina bir-biriga teng bo’lishi mumkin. Har xil o’lchovli matritsalarning bir-biriga teng bo’lishi yoki teng emasligi tushunchalari kiritilmagan. Satrlarining soni ustunlarining soniga teng bo’lgan (𝑚= 𝑛) matritsalar kvadrat matritsalar deyiladi. Agar 𝑖 = 1 bo’lsa, u holda satr-matritsaga ega bo’lamiz; agar
𝑗 = 1 bo’lsa, biz ustun-matritsaga ega bo’lamiz. Ular mos ravishda satr-vektor va
ustun-vektor ham deb ataladi.

Matritsalar va ular ustida amallar

Matritsalar ustidagi asosiy amallarni o’rganamiz.
Matritsalarni qo’shish va ayirish.
Bu amallarni faqat bir xil o’lchovli matritsalar ustida bajarish mumkin. 𝐴 va 𝐵 matritsalarning yig’indisi (ayirmasi) 𝐴 + 𝐵 (𝐴 − 𝐵) bilan belgilanadi. 𝐴 va 𝐵 matritsalarning 𝐴 + 𝐵 (𝐴 − 𝐵) yig’indisi (ayirmasi) deb shunday 𝐶 matritsaga
aytiladiki, 𝐶 matritsaning elementlari 𝑐𝑖 𝑗 = 𝑎𝑖 𝑗 ± 𝑏𝑖 𝑗 dan iboratdir,
bu yerda 𝑎𝑖 𝑗 va 𝑏𝑖 𝑗 - mos ravishda 𝐴 va 𝐵 matritsalarning elementlari.

Matritsalar va ular ustida amallar

Masalan, ikkita

	1	6			−2	4
𝐴 =	2	−4	,	𝐵 =	3	7	.
	−3	9			8	−11

matritsalar berilgan bo’lsin. U holda
𝐴 + 𝐵 =
1 + (−2)
2 + 3
−3 + 8
6 + 4
−4 + 7 =
9 + (−11)
−1 10
5 3
5 −2
,
𝐴 − 𝐵 =
1 − (−2)
2 − 3
−3 − 8
6 − 4
−4 − 7
9 − (−11)
=
3 2
−1 −11
−11 20
.

Matritsalar va ular ustida amallar

Matritsani songa ko’paytirish.
𝐴 matritsani 𝜆 songa ko’paytmasi 𝜆𝐴 bilan belgilanadi.
𝐴 matritsaning 𝜆 songa 𝜆𝐴 ko’paytmasi deb shunday 𝐵 matritsaga aytiladiki, 𝐵
matritsaning elementlari 𝑏𝑖 𝑗 = 𝜆 𝑎𝑖 𝑗 dan iboratdir, bu yerda 𝑎𝑖 𝑗 – 𝐴 matritsaning elementlari. 𝐴 matritsani 𝜆 songa ko’paytirganda hosil bo’ladigan 𝐵 matritsa 𝐴 matritsa bilan bir xil o’lchovli bo’ladi. Hullas, matritsani biror songa ko’paytirish uchun bu matritsaning har bir elementini shu songa ko’paytirib chiqish kerak.

Matritsalar va ular ustida amallar

Masalan,
𝜆 = −2,
𝐴 =
3 0
7 −1
bo’lsin. U holda
𝜆𝐴 = −2 ⋅
3 0
7 −1
=
−6 0
−14 2
.

Matritsalar va ular ustida amallar

Matritsalarni ko’paytirish.
matritsalarning ko’paytmasi deb shunday 𝐶𝑚×𝑝 = 𝐴 ⋅ 𝐵 (sodda qilib,
𝐴𝑚×𝑛 va 𝐵𝑛×𝑝
𝐴𝐵) matritsaga aytiladiki, bu 𝐶 matritsaning elementlari
𝑐 𝑖 𝑗 = 𝑎𝑖1𝑏1𝑗 + 𝑎𝑖 2𝑏2𝑗 + 𝑎𝑖 3𝑏3𝑗 +. . . +𝑎𝑖 𝑛𝑏𝑛𝑗
ko’rinishda bo’ladi, bu yerda 𝑎𝑖 𝑗 va 𝑏𝑖 𝑗 - mos ravishda 𝐴 va 𝐵 matritsalarning
elementlari. Bundan ko’rinadiki, 𝐴 va 𝐵 matritsalarning ko’paytmasi ma’noga ega bo’lishi uchun 𝐴 matritsaning ustunlari soni 𝐵 matritsaning satrlari soniga teng
bo’lishi zarur. Hosil bo’lgan 𝐴𝐵 ko’paytmaning satrlari soni 𝐴 matritsaning satrlari soniga, ustunlari soni esa 𝐵 matritsaning ustunlari soniga teng.

Matritsalar va ular ustida amallar

𝐴𝐵 ko’paytmaning mavjudligidan 𝐵𝐴 ko’paytmaning mavjudligi kelib chiqmaydi. 𝐴𝐵 va 𝐵𝐴 ko’paytmalar mavjud bo’lgan taqdirda ham, odatda (ko’p hollarda), 𝐴𝐵 va 𝐵𝐴 ko’paytmalar bir-biriga teng bo’lmaydi: 𝐴𝐵 ≠ 𝐵𝐴. Agar 𝐴𝐵 = 𝐵𝐴 bo’lsa, u holda 𝐴 va 𝐵 matritsalar o’zaro o’rin almashinuvchi (kommutativ) matritsalar deyiladi.
Ma’lumki, har doim 𝐴𝐵 𝐶 = 𝐴 𝐵𝐶 tenglik o’rinli.

Matritsalar va ular ustida amallar

Misol 1. 𝐴𝐵 va 𝐵𝐴 ko’paytmalarni toping.
𝐴 =
4 −5 8
1 3 −1
,
𝐵 =
−1 5
−2 −3
3 4
.
𝐴𝐵 ko’paytmani topamiz:
𝐴𝐵 =
4 −5 8
1 3 −1
⋅
−1 5
−2 −3
3 4
=
=
4 ⋅ (−1) + (−5) ⋅ (−2) + 8 ⋅ 3
1 ⋅ (−1) + 3 ⋅ (−2) + (−1) ⋅ 3
4 ⋅ 5 + (−5) ⋅ (−3) + 8 ⋅ 4
1 ⋅ 5 + 3 ⋅ (−3) + (−1) ⋅ 4
=
30 67
−10 −8
.

Matritsalar va ular ustida amallar

𝐵𝐴 ko’paytmani topamiz:
𝐵𝐴 =
−1 5
−2 −3
3 4
⋅
4 −5 8
1 3 −1
=
(−1) ⋅ 4 + 5 ⋅ 1
(−2) ⋅ 4 + (−3) ⋅ 1
3 ⋅ 4 + 4 ⋅ 1
(−1) ⋅ (−5) + 5 ⋅ 3
(−2) ⋅ (−5) + (−3) ⋅ 3
3 ⋅ (−5) + 4 ⋅ 3
(−1) ⋅ 8 + 5 ⋅ (−1)
(−2) ⋅ 8 + (−3) ⋅ (−1)
3 ⋅ 8 + 4 ⋅ (−1)
=
1 20 −13
−11 1 −13
16 −3 20
.
Shunday qilib, 𝐴𝐵 ≠ 𝐵𝐴 ekan.

Matritsalar va ular ustida amallar

Misol 2. 𝐴𝐵 va 𝐵𝐴 ko’paytmalarni toping.
𝐴 =
3 5
1 2
, 𝐵 =
1 −5
−1 2
.
Hisoblaymiz:
𝐴𝐵 =
⋅
3 5 1 −5
1 2 −1 2
=
3 ⋅ 1 + 5 ⋅ (−1)
1 ⋅ 1 + 2 ⋅ (−1)
3 ⋅ (−5) + 5 ⋅ 2
1 ⋅ (−5) + 2 ⋅ 2
=
−2 −5
−1 −1
,
𝐵𝐴 =
⋅
1 −5 3 5
−1 2 1 2
=
1 ⋅ 3 + (−5) ⋅ 1
(−1) ⋅ 3 + 2 ⋅ 1
1 ⋅ 5 + (−5) ⋅ 2
(−1) ⋅ 5 + 2 ⋅ 2
=
−2 −5
−1 −1
.
Shunday qilib, 𝐴𝐵 = 𝐵𝐴 ekan.

Matritsalar va ular ustida amallar

Misol 3. 𝐴𝐵 𝐶 va 𝐴 𝐵𝐶 ko’paytmalarni toping.
𝐴 =
1 3
−1 1
2 5
, 𝐵 =
2 −6 1
1 3 −1
, 𝐶 =
−1
2
4
.
Ko’paytmalarni hisoblaymiz:

	5	3	−2			−7
𝐴𝐵 =	−1	9	−2	,	𝐴𝐵 𝐶 =	11	,
	9	3	−3			−15

𝐵𝐶 =
−10
1
, 𝐴 𝐵𝐶 =
−7
11
−15
,
ya`ni
𝐴𝐵 𝐶 = 𝐴 𝐵𝐶 .
E`TIBORINGIZ UCHUN RAHMAT!