|
“Tabiiy fanlar va iqtisodiyot”
kafedrasi assistenti:
Mamaraximova Hulkar Nizomovna
(
2
)
4 ta sondan iborat quyidagi jadvalni qaraymiz
Ushbu
(
1
)
ko`rinishdagi ixtiyoriy
jadvalga
2
-tartibli kvadrat matritsa deb ataymiz
11 12
21
22
a a
A
a a
1-
Ta`rif
.
miqdorga (songa)
(
1
)
-matritsaning ikkinchi tartibli determinanti
deb ataladi. Ta`rifga ko`ra
12
21
22
11
a
a
a
a
11
12
11 22
21 12
21
22
detA=
a
a
a a
a a
a
a
ko`rinishda belgilanadi. Demak, matritsa jadval,
determinant esa, ta`rif bo`yicha sondir.
Belgilashlar:
A
nn
a
,
,
,
12
21
22
11
,
,
,
a
a
a
a
-sonlarga determinantning ele-
mentlari deyiladi
-
1
-satr elementlari,
12
11
,
a
a
21
22
,
a
a
21
11
,
a
a
-
2 –
satr elementlari, -
1
-ustun
elementlari,
12
22
,
a
a
-
2
– elementlari.
22
11
,
a
a
- elementlar joylashgan diagonal
bosh
diagonal
deyiladi.
12
21
,
a
a
diagonal
yordamchi diagonal
deyiladi.
-
elementlar joylashgan
:
i j
i satr
tartibi
a
j ustun tartibi
Misol:
1 2
=1 4
2 3
4
6
2
3 4
(
3
)
.
(
4
)
Uchinchi tartibli determinantlar
9 ta sondan iborat ushbu
11
12
13
21
22
23
31
32
33
a
a
a
A
a
a
a
a
a
a
Jadvalga 3 – tartibli
kvadrat matritsa
deyiladi.
Bu matritsaning determinanti quyidagicha
ifodalangan bo`ladi. Yuqoridagi ta`rifga asosan:
11
12
13
11
12
13
21
22
23
21
22
23
31
32
33
31
32
33
detA=det
=
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
13
22
31
21
12
33
32
23
11
a
a
a
a
a
a
a
a
a
11
22
33
31
12
23
13
21
32
a
a
a
a
a
a
a
a
a
(
5
)
Satr, ustun, bosh va yordamchi diagonallar tushunchasi
2
-tartibli determinantlar kabi kiritiladi
Uchinchi tartibli determinantlarni hisoblash
1-usul
.
Uchburchak usuli
Quyidagi sxemani kiritamiz:
11
12
13
22
23
21
33
31
32
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
a
a
a
a
a
a
a
a
a
13
11
12
22
23
21
31
32
33
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
a
a
a
a
a
a
a
a
a
11
12
13
21
22
23
31
32
33
a
a
a
a
a
a
a
a
a
13
22
31
21
12
33
32
23
11
a a
a
a
a
a
a
a
a
(
5
)
(
6
)
11
22
33
31
12
23
13
21
32
a
a
a
a
a
a
a
a
a
1-
Misol
.
18
0
12
8
0
8
10
1
0
5
)
1
(
4
3
)
2
(
2
2
3
0
)
2
(
2
4
1
5
2
1
5
3
2
4
2
0
2
1
1
11
13
13
12
22
21
11
12
13
11
12
22
21
33
3
11
12
21
22
23
21
22
21
22
3
12
23
2
3
1
32
33
31
32
11
2
31
1
33
3
2
1
2
3
3
3
a
a
a
a a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a a
2-usul. Sarrius usuli:
11
22
33
31
12
23
13
21
32
a
a
a
a
a
a
a
a
a
13
22
31
21
12
33
32
23
11
a
a
a
a
a
a
a
a
a
(
7
)
3-usul. Ixtiyoriy qator elementlari bo`yich yoyib
hisoblash
11
12
13
22 23
21 23
21 22
1 1
1 2
1 3
21
22
23
11
12
13
31 32
32 33
31 33
31
32
33
( 1)
( 1)
( 1)
a
a
a
a a
a a
a a
a
a
a
a
a
a
a a
a a
a a
a
a
a
(
8
)
18
0
12
8
0
8
10
1
0
5
)
1
(
4
3
)
2
(
2
2
3
0
)
2
(
2
4
1
5
2
1
3
2
1
2
0
1
5
3
2
4
2
0
2
1
1
Misol
.
1.
Determinantning satr elementlari bilan ustun
elementlarini
almashtirib
yozish
natijasida
determinantning qiymati o`zgarmaydi.
3. Determinantlarning xossalari
1
2
3
1
1
2
1
0
2
2
0
4
12 8 10 8
18
2
4
5
3
2
5
Misol
.
33
23
13
32
22
12
31
21
11
33
32
31
23
22
21
13
12
11
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
2.
Determinantda 2 tа parallel turgan satr(ustun)i
ning o`rni almashtirilsa determinantning ishorasi
o`zgaradi.
3-
misol
.
1 1
2
0
2
4
-10 +8+8-0+12=18
2
3
5
3.
Determinantning 2 tа satr(ustun)i elementlari
o`zaro teng bo`lsa, uning qiymati nolga teng.
0
15
2
24
15
24
2
1
5
4
3
2
1
3
2
1
4-
misol
4.
Determinantni birorta songa ko`paytirish uchun
uning biror satr(ustun)i elementlarini shu songa
ko`paytirish kifoya.
11
12
13
11
12
13
21
22
23
21
22
23
31
32
33
31
32
33
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
5-
misol
.
2
1
12
6
3
36
3
3
9
27
3
9
27
15
5
16
15
5
16
6-misol
.
1
2
3
3
6
9
0
1
2
7
5.
Determinantning biror satr(ustun)i elementlari
nollardan iborat bo`lsa uning qiymati nolga tengdir
6.
Determinant ikkita satr (ustun)i elementlari mos
ravishda proporsional bo`lsa, uning qiymati nolga
tengdir.
11
1
12
13
11
12
13
1
12
13
21
2
22
23
21
22
23
2
22
33
31
3
32
33
31
32
33
3
32
33
a
b
a
a
a a
a
b a
a
a
b
a
a
a
a
a
b a
a
a
b
a
a
a
a
a
b a
a
.
7.
Determinantning biror satr(ustun)i elementlari
yig`indidan iborat bo`lsa, u holda bu determinant
ikkita determinantning yig`indisiga teng bo`ladi.
11
12
13
11
12
12
13
21
22
23
21
22
22
23
31
32
32
33
31
32
33
a a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
Yuqoridagi xossalar determinantni
hisoblashda qulay
8.
Determinant biror satr (ustun) elementini
ixtiyoriy songa ko`paytirib ixtiyoriy parallel
elementlarini mos ravishda qo`shib yozsak,
determinnatning qiymati o`zgarmaydi.
33
32
31
23
22
21
13
12
11
a
a
a
a
a
a
a
a
a
Algebraik to`ldiruvchi va minorlar.
- determinantni qaraymiz
.
1-
ta`rif.
Determinant biror elementining minori
deb shu determinantdan bu element turgan satr va
ustunni o`chirishdan hosil bo`lgan determinantga
aytiladi.
elementning minori bilan belgilanadi.
ik
a
ik
M
elementning minori deb,
determinantga aytiladi.
elementning minori deb,
determinnatga aytiladi.
33
32
23
22
11
a
a
a
a
M
33
31
23
21
12
a
a
a
a
M
11
a
12
a
element (
1+1=2
,
satr va ustun tartibi
yig`indisi juft
) juft joyda turibdi.
11
a
element (
1+2=3
,
satr va ustun tartibi
yig`indisi toq
) toq joyda turibdi.
12
a
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