Матриц болон тодорхойлогч, тэдгээрийг бодох нь

Ìàòðèö
Òîäîðõîéëîã÷
Óðâóó
ìàòðèö
Ìàòðèö, òîäîðõîéëîã÷,
òýäãýýðèéã õýðýãëýõ íü
Ä.Áàòò°ð
email: batturdn@gmail.com
Îëîí Óëñûí Óëààíáààòàðûí Èõ Ñóðãóóëü
www.iuu.edu.mn
Ä.Áàòò°ð email: batturdn@gmail.com Ìàòåìàòèê 1

Ìàòðèö
Óðâóó
ìàòðèö
ÌÀÒÐÈÖ
"n × m øèðõýã ýëåìåíòýýñ òîãòñîí n øèðõýã ì°ð, m
øèðõýã áàãàíàòàé





a11 a12 ... a1m
a21 a22 ... a2m
...
...
...
...
an1 an2 ... anm





(1)
(1) òýãø °íö°ãò õ³ñíýãòèéã ìàòðèö ãýíý."

Ìàòðèö
Óðâóó
ìàòðèö
ÌÀÒÐÈÖ





a11 a12 ... a1m
a21 a22 ... a2m
...
...
...
...
an1 an2 ... anm





(1)
Æèøýý 1.
A =


−1.5 3
20 5.5
3 −4

 ;

Ìàòðèö
Óðâóó
ìàòðèö
ÌÀÒÐÈÖ





a11 a12 ... a1m
a21 a22 ... a2m
...
...
...
...
an1 an2 ... anm





(1)
Æèøýý 1.
A =


−1.5 3
20 5.5
3 −4

 ;B =


1 0 0
0 1 0
0 0 1

 ;

Ìàòðèö
Óðâóó
ìàòðèö
ÌÀÒÐÈÖ





a11 a12 ... a1m
a21 a22 ... a2m
...
...
...
...
an1 an2 ... anm





(1)
Æèøýý 1.
A =


−1.5 3
20 5.5
3 −4

 ;B =


1 0 0
0 1 0
0 0 1

 ;C = 1 2 5

Ìàòðèö
Óðâóó
ìàòðèö
ÌÀÒÐÈÖ
(1) ìàòðèö: A = [aij] 1≤i≤n; 1≤j≤n (2)
Ì°ðèéí äóãààð: i
Áàãàíûí äóãààð: j
i-ð ì°ð, j−p áàãàíûí îãòëîëöëûí ýëåìåíò: aij

Ìàòðèö
Óðâóó
ìàòðèö
ÌÀÒÐÈÖ
Íýð Õýìæýýñ Æèøýý Òàéëáàð
Ì°ð
ìàòðèö
1 × n 1 2 ... 5 Íýã ì°ð á³õèé
ìàòðèö

Ìàòðèö
Óðâóó
ìàòðèö
ÌÀÒÐÈÖ
Ì°ð
ìàòðèö
1 × n 1 2 ... 5 Íýã ì°ð á³õèé
ìàòðèö
Áàãàíà
ìàòðèö
n × 1





1
2
...
5





Íýã áàãàíà
á³õèé ìàòðèö

Ìàòðèö
Óðâóó
ìàòðèö
ÌÀÒÐÈÖ
Ì°ð
ìàòðèö
1 × n 1 2 ... 5 Íýã ì°ð á³õèé
ìàòðèö
Áàãàíà
ìàòðèö
n × 1





1
2
...
5





Íýã áàãàíà
á³õèé ìàòðèö
Êâàäðàò
ìàòðèö
n × n





1 2 ... 5
2 4 ... 6
...
...
...
...
7 8 ... 9





Ì°ð áà áàãàíû
òîî òýíö³³

Ìàòðèö
Óðâóó
ìàòðèö
ÌÀÒÐÈÖ
Íýãæ
ìàòðèö
n × n





1 0 ... 0
0 1 ... 0
...
...
...
...
0 0 ... 1





äèàãîíàëèéí
äàãóó ç°âõ°í
1, áóñàä
ýëåìåíò³³ä íü
0 áàéõ ìàòðèö

Ìàòðèö
Óðâóó
ìàòðèö
ÌÀÒÐÈÖ
Íýãæ
ìàòðèö
n × n





1 0 ... 0
0 1 ... 0
...
...
...
...
0 0 ... 1





äàãóó ç°âõ°í
1, áóñàä
0 áàéõ ìàòðèö
Ãóðâàëæèí
ìàòðèö
n × n





2 4 ... 8
0 1 ... 10
...
...
...
...
0 0 ... −1





Ãîë
äýýðõ,
ýñâýë äîîðõ
á³ãä 0 áàéõ
ìàòðèöÄ.Áàòò°ð email: batturdn@gmail.com Ìàòåìàòèê 1

Ìàòðèö
Óðâóó
ìàòðèö
ÌÀÒÐÈÖ³íäñýí ³éëäë³³ä
Òîäîðõîéëîëò (Ìàòðèöóóäûã íýìýõ)
Èæèë õýìæýýñ á³õèé

Ìàòðèö
Óðâóó
ìàòðèö
A = [aij] 1≤i≤n; 1≤j≤m
B = [bij] 1≤i≤n; 1≤j≤m

Ìàòðèö
Óðâóó
ìàòðèö
A = [aij] 1≤i≤n; 1≤j≤m
B = [bij] 1≤i≤n; 1≤j≤m
ìàòðèöóóäûí íèéëáýð C = A + B ìàòðèöûã

Ìàòðèö
Óðâóó
ìàòðèö
A = [aij] 1≤i≤n; 1≤j≤m
B = [bij] 1≤i≤n; 1≤j≤m
C = [cij = aij + bij] 1≤i≤n; 1≤j≤m
ãýæ îëíî.

Ìàòðèö
Óðâóó
ìàòðèö
A = [aij] 1≤i≤n; 1≤j≤m
B = [bij] 1≤i≤n; 1≤j≤m
ãýæ îëíî.
Æèøýý 2.
1 3 1
1 0 0
+
0 0 5
7 5 0

Ìàòðèö
Óðâóó
ìàòðèö
A = [aij] 1≤i≤n; 1≤j≤m
B = [bij] 1≤i≤n; 1≤j≤m
ãýæ îëíî.
Æèøýý 2.
1 3 1
1 0 0
+
0 0 5
7 5 0
=
1 + 0 3 + 0 1 + 5
1 + 7 0 + 5 0 + 0

Ìàòðèö
Óðâóó
ìàòðèö
A = [aij] 1≤i≤n; 1≤j≤m
B = [bij] 1≤i≤n; 1≤j≤m
ãýæ îëíî.
Æèøýý 2.
1 3 1
1 0 0
+
0 0 5
7 5 0
=
1 + 0 3 + 0 1 + 5
1 + 7 0 + 5 0 + 0
=
1 3 6
8 5 0

Ìàòðèö
Óðâóó
ìàòðèö
Òîäîðõîéëîëò (Ìàòðèöûã òîîãîîð ³ðæ³³ëýõ)
A = [aij] 1≤i≤n; 1≤j≤m ìàòðèöûã c ∈ R òîãòìîëîîð ³ðæ³³ëñýí
³ðæâýð ìàòðèöûã

Ìàòðèö
Óðâóó
ìàòðèö
c · A = [c · aij] 1≤i≤n; 1≤j≤m
ãýæ îëíî.

Ìàòðèö
Óðâóó
ìàòðèö
c · A = [c · aij] 1≤i≤n; 1≤j≤m
ãýæ îëíî.
Æèøýý 3.
2 ·
1 3 1
1 0 0
=
=
2 · 1 2 · 3 2 · 1
2 · 1 2 · 0 2 · 0
=
2 6 2
2 0 0

Ìàòðèö
Óðâóó
ìàòðèö
Òîäîðõîéëîëò (Ìàòðèöûã õ°ðâ³³ëýõ)
A = [aij] 1≤i≤n; 1≤j≤m ìàòðèöûí õ°ðâ³³ëñýí ìàòðèö AT
ìàòðèöûã

Ìàòðèö
Óðâóó
ìàòðèö
ìàòðèöûã
AT
= [aT
ji = aij] 1≤j≤m; 1≤i≤n
ãýæ îëíî.

Ìàòðèö
Óðâóó
ìàòðèö
ìàòðèöûã
AT
= [aT
ãýæ îëíî.
Æèøýý 4.
1 2 3
0 −6 7
T
=

Ìàòðèö
Óðâóó
ìàòðèö
ìàòðèöûã
AT
= [aT
ãýæ îëíî.
Æèøýý 4.
1 2 3
0 −6 7
T
=
=


1 0

Ìàòðèö
Óðâóó
ìàòðèö
ìàòðèöûã
AT
= [aT
ãýæ îëíî.
Æèøýý 4.
1 2 3
0 −6 7
T
=
=


1 0
2 −6

Ìàòðèö
Óðâóó
ìàòðèö
ìàòðèöûã
AT
= [aT
ãýæ îëíî.
Æèøýý 4.
1 2 3
0 −6 7
T
=
=


1 0
2 −6
3 7



Ìàòðèö
Óðâóó
ìàòðèö
Òîäîðõîéëîëò (Ìàòðèöûã ìàòðèöààð ³ðæ³³ëýõ)
A = [aij] 1≤i≤n; 1≤j≤m
B = [bij] 1≤i≤m; 1≤j≤k
ìàòðèöóóäûí ³ðæâýð ìàòðèö C = A · B ìàòðèöûã

Ìàòðèö
Óðâóó
ìàòðèö
Òîäîðõîéëîëò (Ìàòðèöûã ìàòðèöààð ³ðæ³³ëýõ)
A = [aij] 1≤i≤n; 1≤j≤m
B = [bij] 1≤i≤m; 1≤j≤k
ìàòðèöóóäûí ³ðæâýð ìàòðèö C = A · B ìàòðèöûã
C = [cij = ai1·b1j+ai2·b2j+· · ·+aim·bmj =
m
r=1
air·brj] 1≤i≤n; 1≤j≤k
ãýæ îëíî.

Ìàòðèö
Óðâóó
ìàòðèö
Çóðàã: show an example picture

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 5.
1 2 3
0 − 6 7
∗

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 5.
1 2 3
0 − 6 7
∗


1 0
2 − 6
3 7

 =

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 5.
1 2 3
0 − 6 7
∗


1 0
2 − 6
3 7

 =
1 ∗ 1 + 2 ∗ 2 + 3 ∗ 3

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 5.
1 2 3
0 − 6 7
∗


1 0
2 − 6
3 7

 =
1 ∗ 1 + 2 ∗ 2 + 3 ∗ 3 1 ∗ 0 + 2 ∗ (−6) + 3 ∗ 7

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 5.
1 2 3
0 − 6 7
∗


1 0
2 − 6
3 7

 =
1 ∗ 1 + 2 ∗ 2 + 3 ∗ 3 1 ∗ 0 + 2 ∗ (−6) + 3 ∗ 7
0 ∗ 1 + (−6) ∗ 2 + 7 ∗ 3

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 5.
1 2 3
0 − 6 7
∗


1 0
2 − 6
3 7

 =
1 ∗ 1 + 2 ∗ 2 + 3 ∗ 3 1 ∗ 0 + 2 ∗ (−6) + 3 ∗ 7
0 ∗ 1 + (−6) ∗ 2 + 7 ∗ 3 0 ∗ 0 + (−6) ∗ (−6) + 7 ∗ 7

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 5.
1 2 3
0 − 6 7
∗


1 0
2 − 6
3 7

 =
1 ∗ 1 + 2 ∗ 2 + 3 ∗ 3 1 ∗ 0 + 2 ∗ (−6) + 3 ∗ 7
0 ∗ 1 + (−6) ∗ 2 + 7 ∗ 3 0 ∗ 0 + (−6) ∗ (−6) + 7 ∗ 7
=
14 9
9 84

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 5.
1 2 3
0 − 6 7
∗


1 0
2 − 6
3 7

 =
1 ∗ 1 + 2 ∗ 2 + 3 ∗ 3 1 ∗ 0 + 2 ∗ (−6) + 3 ∗ 7
0 ∗ 1 + (−6) ∗ 2 + 7 ∗ 3 0 ∗ 0 + (−6) ∗ (−6) + 7 ∗ 7
=
14 9
9 84
=
14 9
9 84

Ìàòðèö
Óðâóó
ìàòðèö
ÌÀÒÐÈÖ³íäñýí ³éëäë³³äèéí ÷àíàðóóä
Ìàòðèöûí íýìýõ
1 A + B = B + A
2 (A + B) + C = A + (B + C)
Ìàòðèöûã òîîãîîð ³ðæ³³ëýõ
1 (a + b) · A = a · A + b · A
2 (a · b) · A = a · (b · A)
3 a · (A + B) = a · A + a · B
4 0 · A = 0; 1 · A = A

Ìàòðèö
Óðâóó
ìàòðèö
ÌÀÒÐÈÖ³íäñýí ³éëäë³³äèéí ÷àíàðóóä
Ìàòðèöûí ³ðæ³³ëýõ
1 (A · B) · C = A · (B · C)
2 A · B = B · A
3 (A + B) · C = A · C + B · C
4 A · (B + C) = A · B + A · C
5 A · E = E · A = A
6 A · 0 = 0 · A = 0
7 (A · B)T = BT · AT

Ìàòðèö
Óðâóó
ìàòðèö
ÒÎÄÎÐÕÎÉËÎÃ×
"n × n øèðõýã ýëåìåíòýýñ òîãòñîí n øèðõýã ì°ð, n
detA =
a11 a12 ... a1n
a21 a22 ... a2n
...
...
...
...
an1 an2 ... ann
(2)
(2) êâàäðàò õ³ñíýãòèéã n-ð ýðýìáèéí A ìàòðèöûí
òîäîðõîéëîã÷ ãýõ ãýíý."

Ìàòðèö
Óðâóó
ìàòðèö
detA =
a11 a12 ... a1n
a21 a22 ... a2n
...
...
...
...
an1 an2 ... ann
(2)
Æèøýý 6:
11 2
−3 1
;

Ìàòðèö
Óðâóó
ìàòðèö
detA =
a11 a12 ... a1n
a21 a22 ... a2n
...
...
...
...
an1 an2 ... ann
(2)
Æèøýý 6:
11 2
−3 1
;
1 2 0
−1 −3 1
1 −3 1
;

Ìàòðèö
Óðâóó
ìàòðèö
detA =
a11 a12 ... a1n
a21 a22 ... a2n
...
...
...
...
an1 an2 ... ann
(2)
Æèøýý 6:
11 2
−3 1
;
1 2 0
−1 −3 1
1 −3 1
;
1 2 0 5
3 −1 −3 1
0 1 −3 1
6 −3 3 1

Ìàòðèö
Óðâóó
ìàòðèö
ÒÎÄÎÐÕÎÉËÎÃ×2,3-ð ýðýìáèéí òîäîðõîéëîã÷èéã áîäîõ
detA =
a11 a12
a21 a22

Ìàòðèö
Óðâóó
ìàòðèö
detA =
a11 a12
a21 a22
= a11a22 − a12a21

Ìàòðèö
Óðâóó
ìàòðèö
detA =
a11 a12
a21 a22
= a11a22 − a12a21
detB =
b11 b12 b13
b21 b22 b23
b31 b32 b33

Ìàòðèö
Óðâóó
ìàòðèö
detA =
a11 a12
a21 a22
= a11a22 − a12a21
detB =
b11 b12 b13
b21 b22 b23
b31 b32 b33
= b11b22b33

Ìàòðèö
Óðâóó
ìàòðèö
detA =
a11 a12
a21 a22
= a11a22 − a12a21
detB =
b11 b12 b13
b21 b22 b23
b31 b32 b33
= b11b22b33 + b13b21b32

Ìàòðèö
Óðâóó
ìàòðèö
detA =
a11 a12
a21 a22
= a11a22 − a12a21
detB =
b11 b12 b13
b21 b22 b23
b31 b32 b33
= b11b22b33 + b13b21b32 +
b12b23b31

Ìàòðèö
Óðâóó
ìàòðèö
detA =
a11 a12
a21 a22
= a11a22 − a12a21
detB =
b11 b12 b13
b21 b22 b23
b31 b32 b33
= b11b22b33 + b13b21b32 +
b12b23b31 − b13b22b31

Ìàòðèö
Óðâóó
ìàòðèö
detA =
a11 a12
a21 a22
= a11a22 − a12a21
detB =
b11 b12 b13
b21 b22 b23
b31 b32 b33
= b11b22b33 + b13b21b32 +
b12b23b31 − b13b22b31 − b11b23b32

Ìàòðèö
Óðâóó
ìàòðèö
detA =
a11 a12
a21 a22
= a11a22 − a12a21
detB =
b11 b12 b13
b21 b22 b23
b31 b32 b33
= b11b22b33 + b13b21b32 +
b12b23b31 − b13b22b31 − b11b23b32 − b12b21b33

Ìàòðèö
Óðâóó
ìàòðèö
detA =
a11 a12
a21 a22
= a11a22 − a12a21
detB =
b11 b12 b13
b21 b22 b23
b31 b32 b33
= b11b22b33 + b13b21b32 +
b12b23b31 − b13b22b31 − b11b23b32 − b12b21b33
Æèøýý 7:
11 2
−3 1

Ìàòðèö
Óðâóó
ìàòðèö
detA =
a11 a12
a21 a22
= a11a22 − a12a21
detB =
b11 b12 b13
b21 b22 b23
b31 b32 b33
= b11b22b33 + b13b21b32 +
b12b23b31 − b13b22b31 − b11b23b32 − b12b21b33
Æèøýý 7:
11 2
−3 1
= 11 · 1 − 2 · (−3)

Ìàòðèö
Óðâóó
ìàòðèö
detA =
a11 a12
a21 a22
= a11a22 − a12a21
detB =
b11 b12 b13
b21 b22 b23
b31 b32 b33
= b11b22b33 + b13b21b32 +
b12b23b31 − b13b22b31 − b11b23b32 − b12b21b33
Æèøýý 7:
11 2
−3 1
= 11 · 1 − 2 · (−3) = 17

Ìàòðèö
Óðâóó
ìàòðèö
detA =
a11 a12
a21 a22
= a11a22 − a12a21
detB =
b11 b12 b13
b21 b22 b23
b31 b32 b33
= b11b22b33 + b13b21b32 +
b12b23b31 − b13b22b31 − b11b23b32 − b12b21b33
Æèøýý 7:
11 2
−3 1
= 11 · 1 − 2 · (−3) = 17
1 2 0
−1 −3 1
1 −3 1

Ìàòðèö
Óðâóó
ìàòðèö
detA =
a11 a12
a21 a22
= a11a22 − a12a21
detB =
b11 b12 b13
b21 b22 b23
b31 b32 b33
= b11b22b33 + b13b21b32 +
b12b23b31 − b13b22b31 − b11b23b32 − b12b21b33
Æèøýý 7:
11 2
−3 1
= 11 · 1 − 2 · (−3) = 17
1 2 0
−1 −3 1
1 −3 1
= 1 · (−3) · 1 + 0 · (−1) · (−3) + 2 · 1 · 1

Ìàòðèö
Óðâóó
ìàòðèö
detA =
a11 a12
a21 a22
= a11a22 − a12a21
detB =
b11 b12 b13
b21 b22 b23
b31 b32 b33
= b11b22b33 + b13b21b32 +
b12b23b31 − b13b22b31 − b11b23b32 − b12b21b33
Æèøýý 7:
11 2
−3 1
= 11 · 1 − 2 · (−3) = 17
1 2 0
−1 −3 1
1 −3 1
= 1 · (−3) · 1 + 0 · (−1) · (−3) + 2 · 1 · 1
−0 · (−3) · 1 − 1 · 1 · (−3) − 2 · (−1) · 1

Ìàòðèö
Óðâóó
ìàòðèö
detA =
a11 a12
a21 a22
= a11a22 − a12a21
detB =
b11 b12 b13
b21 b22 b23
b31 b32 b33
= b11b22b33 + b13b21b32 +
b12b23b31 − b13b22b31 − b11b23b32 − b12b21b33
Æèøýý 7:
11 2
−3 1
= 11 · 1 − 2 · (−3) = 17
1 2 0
−1 −3 1
1 −3 1
= 1 · (−3) · 1 + 0 · (−1) · (−3) + 2 · 1 · 1
−0 · (−3) · 1 − 1 · 1 · (−3) − 2 · (−1) · 1 =

Ìàòðèö
Óðâóó
ìàòðèö
ÒÎÄÎÐÕÎÉËÎÃ×Òîäîðõîéëîã÷èéí ÷àíàðóóä
1. Òîäîðõîéëîã÷èéã õ°ðâ³³ëýõýä (á³õ ì°ðèéã õàðãàëçàõ
äóãààðòàé áàãàíààð, á³õ áàãàíûí õàðãàëçàõ äóãààðòàé
ì°ð°°ð ñîëèõîä) òîäîðõîéëîã÷èéí óòãà °°ð÷ë°ãä°õã³é.
a1 b1 c1
a2 b2 c2
a3 b3 c3
=
a1 a2 a3
b1 b2 b3
c1 c2 c3

Ìàòðèö
Óðâóó
ìàòðèö
1. Òîäîðõîéëîã÷èéã õ°ðâ³³ëýõýä (á³õ ì°ðèéã õàðãàëçàõ
äóãààðòàé áàãàíààð, á³õ áàãàíûí õàðãàëçàõ äóãààðòàé
ì°ð°°ð ñîëèõîä) òîäîðõîéëîã÷èéí óòãà °°ð÷ë°ãä°õã³é.
a1 b1 c1
a2 b2 c2
a3 b3 c3
=
a1 a2 a3
b1 b2 b3
c1 c2 c3
2. Òîäîðõîéëîã÷èéí äóðûí õî¼ð ì°ð (áàãàíà)-èéí áàéðûã
ñîëèõîä òîäîðõîéëîã÷èéí òýìäýã ýñðýãýýð °°ð÷ë°ãä°í°.
a1 b1 c1
a2 b2 c2
a3 b3 c3
= −
b1 a1 c1
b2 a2 c2
b3 a3 c3

Ìàòðèö
Óðâóó
ìàòðèö
3. Òîäîðõîéëîã÷ õî¼ð èæèë ì°ð (áàãàíà)-òýé áîë
òîäîðõîéëîã÷èéí óòãà 0 áàéíà.

Ìàòðèö
Óðâóó
ìàòðèö
a1 b1 c1
a1 b1 c1
a3 b3 c3
=

Ìàòðèö
Óðâóó
ìàòðèö
a1 b1 c1
a1 b1 c1
a3 b3 c3
=
ãýâýë ÷àíàð 2-ð
a1 b1 c1
a1 b1 c1
a3 b3 c3
= −
áîëíî. Èéìä = − áóþó 2 = 0 ãýäãýýñ = 0
áàéíà.

Ìàòðèö
Óðâóó
ìàòðèö
4. Òîäîðõîéëîã÷èéí àëü íýã ì°ð (áàãàíà)-íä åð°íõèé
³ðæèãäõ³³íèéã òîäîðõîéëîã÷èéí òýìäýãèéí °ìí°
ãàðãàæ áîëíî.
a1 mb1 c1
a2 mb2 c2
a3 mb3 c3
= m ·
a1 b1 c1
a2 b2 c2
a3 b3 c3

Ìàòðèö
Óðâóó
ìàòðèö
a1 mb1 c1
a2 mb2 c2
a3 mb3 c3
= m ·
a1 b1 c1
a2 b2 c2
a3 b3 c3
Ì°ðäë°ã 1:
Õýðýâ òîäîðõîéëîã÷èéí àëü íýã ì°ðèéí (áàãàíà)-°íä á³õ ýëåìåíò
0 áàéâàë òîäîðõîéëîã÷èéí óòãà 0 áàéíà.

Ìàòðèö
Óðâóó
ìàòðèö
a1 mb1 c1
a2 mb2 c2
a3 mb3 c3
= m ·
a1 b1 c1
a2 b2 c2
a3 b3 c3
Ì°ðäë°ã 1:
Õýðýâ òîäîðõîéëîã÷èéí àëü íýã ì°ðèéí (áàãàíà)-°íä á³õ ýëåìåíò
0 áàéâàë òîäîðõîéëîã÷èéí óòãà 0 áàéíà.
Ì°ðäë°ã 2:
Õýðýâ òîäîðõîéëîã÷èéí õî¼ð ì°ð (áàãàíà)-èéí ýëåìåíò³³ä
õàðãàëçàí ïðîïîðöèîíàëü áîë òîäîðõîéëîã÷èéí óòãà 0 áàéíà.

Ìàòðèö
Óðâóó
ìàòðèö
5. Òîäîðõîéëîã÷èéí àëü íýã áàãàíà (ì°ð) -èéí á³õ
ýëåìåíò õî¼ð ýëåìåíòèéí íèéëáýðò áè÷èæ áîëîõ áà
a1 + a1 b1 c1
a2 + a2 b2 c2
a3 + a3 b3 c3
=
a1 b1 c1
a2 b2 c2
a3 b3 c3
+
a1 b1 c1
a2 b2 c2
a3 b3 c3
àäèëòãàë áèåëýíý.

Ìàòðèö
Óðâóó
ìàòðèö
6. Òîäîðõîéëîã÷èéí äóðûí ì°ð (áàãàíà)-èéã äóðûí
òîãòìîëîîð ³ðæ³³ëæ °°ð ì°ð (áàãàíà) äýýð íýìýõ
(õàñàõ)-ä òîäîðõîéëîã÷èéí óòãà °°ð÷ë°ãä°õã³é.
a1 b1 c1
a2 b2 c2
a3 b3 c3
=
a1 b1 + ka1 c1
a2 b2 + ka2 c2
a3 b3 + ka3 c3

Ìàòðèö
Óðâóó
ìàòðèö
a1 b1 c1
a2 b2 c2
a3 b3 c3
=
a1 b1 + ka1 c1
a2 b2 + ka2 c2
a3 b3 + ka3 c3
Æèøýý 8:
1 2 0
0 2 4
7 1 3

Ìàòðèö
Óðâóó
ìàòðèö
a1 b1 c1
a2 b2 c2
a3 b3 c3
=
a1 b1 + ka1 c1
a2 b2 + ka2 c2
a3 b3 + ka3 c3
Æèøýý 8:
1 2 0
0 2 4
7 1 3
=
1 2 0
0 + 2 + 4 +
7 1 3

Ìàòðèö
Óðâóó
ìàòðèö
a1 b1 c1
a2 b2 c2
a3 b3 c3
=
a1 b1 + ka1 c1
a2 b2 + ka2 c2
a3 b3 + ka3 c3
Æèøýý 8:
1 2 0
0 2 4
7 1 3
=
1 2 0
0 + · 1 2 + · 2 4 + · 0
7 1 3

Ìàòðèö
Óðâóó
ìàòðèö
a1 b1 c1
a2 b2 c2
a3 b3 c3
=
a1 b1 + ka1 c1
a2 b2 + ka2 c2
a3 b3 + ka3 c3
Æèøýý 8:
1 2 0
0 2 4
7 1 3
=
1 2 0
0 + · 1 2 + · 2 4 + · 0
7 1 3
=
1 2 0
7 1 3

Ìàòðèö
Óðâóó
ìàòðèö
ÒÎÄÎÐÕÎÉËÎÃ×Äýýä ýðýìáèéí òîäîðõîéëîã÷èéã áîäîõ
1. Òîäîðõîéëîã÷èéã ì°ð (áàãàíà)-°°ð çàäëàõ:
n-ð ýðýìáèéí òîäîðõîéëîã÷èéí i-ð ì°ð, -ð ì°ð áàãàíûã
äàðàõàä ³³ñýõ n − 1-ð ýðýìáèéí òîäîðõîéëîã÷èéã
äàðàãäñàí ì°ð áàãàíû îãòëîëöîë äýýðõè aij ýëåìåíòýä
õàðãàëçàõ ìèíîð ãýýä Mij ãýæ òýìäýãëýíý.

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 9:
detA =
1 2 3
4 5 6
7 8 9
òîäîðõîéëîã÷èéí õóâüä:

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 9:
detA =
1 2 3
4 5 6
7 8 9
M13 =
4 5
7 8
;

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 9:
detA =
1 2 3
4 5 6
7 8 9
M13 =
4 5
7 8
; M32 =
1 3
4 6
;

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 9:
detA =
1 2 3
4 5 6
7 8 9
M13 =
4 5
7 8
; M32 =
1 3
4 6
;
M21 = ;

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 9:
detA =
1 2 3
4 5 6
7 8 9
M13 =
4 5
7 8
; M32 =
1 3
4 6
;
M21 = ; M = ;

Ìàòðèö
Óðâóó
ìàòðèö
aij ýëåìåíòýä õàðãàëçàõ ìèíîð Mij-ã (−1)i+j òýìäýãýýð
³ðæ³³ëñýí ³ðæâýðèéã ýíý ýëåìåíòýä õàðãàëçàõ àëãåáðûí
ã³éöýýëò ãýýä Aij ãýæ òýìäýãëýíý.

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 9 (³ðãýëæëýë):
detA =
1 2 3
4 5 6
7 8 9

Ìàòðèö
Óðâóó
ìàòðèö
detA =
1 2 3
4 5 6
7 8 9
A13

Ìàòðèö
Óðâóó
ìàòðèö
detA =
1 2 3
4 5 6
7 8 9
A13 = (−1)1+3 4 5
7 8

Ìàòðèö
Óðâóó
ìàòðèö
detA =
1 2 3
4 5 6
7 8 9
A13 = (−1)1+3 4 5
7 8
=
4 5
7 8

Ìàòðèö
Óðâóó
ìàòðèö
detA =
1 2 3
4 5 6
7 8 9
A13 = (−1)1+3 4 5
7 8
=
4 5
7 8
A32

Ìàòðèö
Óðâóó
ìàòðèö
detA =
1 2 3
4 5 6
7 8 9
A13 = (−1)1+3 4 5
7 8
=
4 5
7 8
A32 = (−1)3+2 1 3
4 6

Ìàòðèö
Óðâóó
ìàòðèö
detA =
1 2 3
4 5 6
7 8 9
A13 = (−1)1+3 4 5
7 8
=
4 5
7 8
A32 = (−1)3+2 1 3
4 6
= −
1 3
4 6

Ìàòðèö
Óðâóó
ìàòðèö
detA =
1 2 3
4 5 6
7 8 9
A13 = (−1)1+3 4 5
7 8
=
4 5
7 8
A32 = (−1)3+2 1 3
4 6
= −
1 3
4 6
A

Ìàòðèö
Óðâóó
ìàòðèö
detA =
1 2 3
4 5 6
7 8 9
A13 = (−1)1+3 4 5
7 8
=
4 5
7 8
A32 = (−1)3+2 1 3
4 6
= −
1 3
4 6
A = (−1)

Ìàòðèö
Óðâóó
ìàòðèö
detA =
1 2 3
4 5 6
7 8 9
A13 = (−1)1+3 4 5
7 8
=
4 5
7 8
A32 = (−1)3+2 1 3
4 6
= −
1 3
4 6
A = (−1) =

Ìàòðèö
Óðâóó
ìàòðèö
Theorem
Òîäîðõîéëîã÷èéí óòãà äóðûí ì°ð (áàãàíà)-ûí ýëåìåíò
á³ðèéã ò³³íèé àëãåáðûí ã³éöýýëòýýð ³ðæ³³ëæ íýìñýí
íèéëáýðòýé òýíö³³.
detA = ai1 · Ai1 + ai2 · Ai2 + · · · + ain · Ain, i = 1, 2, . . . , n
detA = a1j · A1j + a2j · A2j + · · · + anj · Anj, j = 1, 2, . . . , n
(3)

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 9:
detA =
1 2 3
4 5 6
7 8 9
=

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 9:
detA =
1 2 3
4 5 6
7 8 9
= a21A21 + a22A22 + a23A23
=

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 9:
detA =
1 2 3
4 5 6
7 8 9
= a21A21 + a22A22 + a23A23
= 4(−1)2+1 2 3
8 9
+

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 9:
detA =
1 2 3
4 5 6
7 8 9
= a21A21 + a22A22 + a23A23
= 4(−1)2+1 2 3
8 9
+5(−1)2+2 1 3
7 9
+

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 9:
detA =
1 2 3
4 5 6
7 8 9
= a21A21 + a22A22 + a23A23
= 4(−1)2+1 2 3
8 9
+5(−1)2+2 1 3
7 9
+6(−1)2+3 1 2
7 8

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 9:
detA =
1 2 3
4 5 6
7 8 9
= a21A21 + a22A22 + a23A23
= 4(−1)2+1 2 3
8 9
+5(−1)2+2 1 3
7 9
+6(−1)2+3 1 2
7 8
= 4(−1)(2 · 9 − 3 · 8)

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 9:
detA =
1 2 3
4 5 6
7 8 9
= a21A21 + a22A22 + a23A23
= 4(−1)2+1 2 3
8 9
+5(−1)2+2 1 3
7 9
+6(−1)2+3 1 2
7 8
= 4(−1)(2 · 9 − 3 · 8) + 5(1 · 9 − 3 · 7) +

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 9:
detA =
1 2 3
4 5 6
7 8 9
= a21A21 + a22A22 + a23A23
= 4(−1)2+1 2 3
8 9
+5(−1)2+2 1 3
7 9
+6(−1)2+3 1 2
7 8
= 4(−1)(2 · 9 − 3 · 8) + 5(1 · 9 − 3 · 7) + 6(−1)(1 · 8 − 2 · 7)

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 9:
detA =
1 2 3
4 5 6
7 8 9
= a21A21 + a22A22 + a23A23
= 4(−1)2+1 2 3
8 9
+5(−1)2+2 1 3
7 9
+6(−1)2+3 1 2
7 8
= 4(−1)(2 · 9 − 3 · 8) + 5(1 · 9 − 3 · 7) + 6(−1)(1 · 8 − 2 · 7)
= −4( ) +

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 9:
detA =
1 2 3
4 5 6
7 8 9
= a21A21 + a22A22 + a23A23
= 4(−1)2+1 2 3
8 9
+5(−1)2+2 1 3
7 9
+6(−1)2+3 1 2
7 8
= 4(−1)(2 · 9 − 3 · 8) + 5(1 · 9 − 3 · 7) + 6(−1)(1 · 8 − 2 · 7)
= −4( ) + 5( ) −

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 9:
detA =
1 2 3
4 5 6
7 8 9
= a21A21 + a22A22 + a23A23
= 4(−1)2+1 2 3
8 9
+5(−1)2+2 1 3
7 9
+6(−1)2+3 1 2
7 8
= 4(−1)(2 · 9 − 3 · 8) + 5(1 · 9 − 3 · 7) + 6(−1)(1 · 8 − 2 · 7)
= −4( ) + 5( ) − 6( ) =

Ìàòðèö
Óðâóó
ìàòðèö
ÌàòðèöÓðâóó ìàòðèöûã îëîõ
Òîäîðõîéëîëò
An×n ìàòðèöûí òîäîðõîéëîã÷ òýãòýé òýíö³³ (|A| = 0) áîë á°õ°õ
ìàòðèö, ýñðýã òîõèîëäîëä (|A| = 0) ³ë á°õ°õ ìàòðèö ãýíý.
Õýðýâ ³ë á°õ°õ An×n ìàòðèöûí õóâüä A · B = B · A = E áàéõ
Bn×n ìàòðèöûã A ìàòðèöûí óðâóó ìàòðèö ãýýä A−1
ãýæ
òýìäýãëýíý.

Ìàòðèö
Óðâóó
ìàòðèö
ÌàòðèöÓðâóó ìàòðèöûã îëîõ
An×n ìàòðèöûí òîäîðõîéëîã÷ òýãòýé òýíö³³ (|A| = 0) áîë á°õ°õ
ìàòðèö, ýñðýã òîõèîëäîëä (|A| = 0) ³ë á°õ°õ ìàòðèö ãýíý.
Õýðýâ ³ë á°õ°õ An×n ìàòðèöûí õóâüä A · B = B · A = E áàéõ
Bn×n ìàòðèöûã A ìàòðèöûí óðâóó ìàòðèö ãýýä A−1
ãýæ
òýìäýãëýíý.
Theorem
Õýðýâ An×n ìàòðèö ³ë á°õ°õ áîë öîðûí ãàíö óðâóó ìàòðèö A−1
îðøèí áàéõ áà ò³³íèéã
A−1
=
1
|A|
(Av
)T
ãýæ îëíî. Ýíä Av
íü A ìàòðèöûí àëãåáðûí ã³éöýýëò³³äýýñ
çîõèîñîí ìàòðèö.Ä.Áàòò°ð email: batturdn@gmail.com Ìàòåìàòèê 1

Ìàòðèö
Óðâóó
ìàòðèö
ÌàòðèöÓðâóó ìàòðèöûã îëîõ (àëãåáðûí ã³éöýýëò àøèãëàõ àðãà)
1 ã°ãäñ°í A ìàòðèöûí òîäîðõîéëîã÷èéã áîäíî. /Õýðýâ
òîäîðõîéëîã÷ íü 0 áîë óðâóó ìàòðèö îðøèõã³é, 0-ýýñ
ÿëãààòàé áîë óðâóó ìàòðèö îðøèí áàéíà.
2 ã°ãäñ°í A ìàòðèöûí á³õ àëãåáðûí ã³éöýýëò³³äèéã îëíî.
3 Àëãåáðûí ã³éöýýëò³³äýýð Av
ìàòðèö ³³ñãýíý.
4 “ññýí àëãåáðûí ã³éöýýëò³³äýñ òîãòîõ Av
ìàòðèöûã
õ°ðâ³³ëíý.
5 Õ°ðâ³³ëñýí (Av
)T
ìàòðèöûí ýëåìåíò á³ðèéã °ã°ãäñ°í
ìàòðèöûí òîäîðõîéëîã÷èä õóâààñíààð óðâóó ìàòðèö A−1
îëäíî.

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 10:
A =
1 2
3 4
ìàòðèöûí óðâóó ìàòðèöûã îë.

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 10:
A =
1 2
3 4
1 |A| =
1 2
3 4
= 1 · 4 − 2 · 3 = 4 − 6 = −2 = 0 òóë óðâóó
ìàòðèö îðøèí áàéíà.

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 10:
A =
1 2
3 4
1 |A| =
1 2
3 4
= 1 · 4 − 2 · 3 = 4 − 6 = −2 = 0 òóë óðâóó
2 A11 = (−1)1+1
· 4 = 4,

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 10:
A =
1 2
3 4
1 |A| =
1 2
3 4
= 1 · 4 − 2 · 3 = 4 − 6 = −2 = 0 òóë óðâóó
2 A11 = (−1)1+1
· 4 = 4, A12 = (−1)1+2
· 3 = −3

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 10:
A =
1 2
3 4
1 |A| =
1 2
3 4
= 1 · 4 − 2 · 3 = 4 − 6 = −2 = 0 òóë óðâóó
2 A11 = (−1)1+1
· 4 = 4, A12 = (−1)1+2
· 3 = −3
A21
= (−1)2+1
· 2 = −2,

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 10:
A =
1 2
3 4
1 |A| =
1 2
3 4
= 1 · 4 − 2 · 3 = 4 − 6 = −2 = 0 òóë óðâóó
2 A11 = (−1)1+1
· 4 = 4, A12 = (−1)1+2
· 3 = −3
A21
= (−1)2+1
· 2 = −2, A22 = (−1)2+2
· 1 = 1

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 10:
A =
1 2
3 4
1 |A| =
1 2
3 4
= 1 · 4 − 2 · 3 = 4 − 6 = −2 = 0 òóë óðâóó
2 A11 = (−1)1+1
· 4 = 4, A12 = (−1)1+2
· 3 = −3
A21
= (−1)2+1
· 2 = −2, A22 = (−1)2+2
· 1 = 1
3 Av
=
4 −3
−2 1
⇒

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 10:
A =
1 2
3 4
1 |A| =
1 2
3 4
= 1 · 4 − 2 · 3 = 4 − 6 = −2 = 0 òóë óðâóó
2 A11 = (−1)1+1
· 4 = 4, A12 = (−1)1+2
· 3 = −3
A21
= (−1)2+1
· 2 = −2, A22 = (−1)2+2
· 1 = 1
3 Av
=
4 −3
−2 1
⇒
4 (Av
)T
=
4 −3
−2 1
T
=

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 10:
A =
1 2
3 4
1 |A| =
1 2
3 4
= 1 · 4 − 2 · 3 = 4 − 6 = −2 = 0 òóë óðâóó
2 A11 = (−1)1+1
· 4 = 4, A12 = (−1)1+2
· 3 = −3
A21
= (−1)2+1
· 2 = −2, A22 = (−1)2+2
· 1 = 1
3 Av
=
4 −3
−2 1
⇒
4 (Av
)T
=
4 −3
−2 1
T
=
4 −2
−3 1

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 10:
A =
1 2
3 4
1 |A| =
1 2
3 4
= 1 · 4 − 2 · 3 = 4 − 6 = −2 = 0 òóë óðâóó
2 A11 = (−1)1+1
· 4 = 4, A12 = (−1)1+2
· 3 = −3
A21
= (−1)2+1
· 2 = −2, A22 = (−1)2+2
· 1 = 1
3 Av
=
4 −3
−2 1
⇒
4 (Av
)T
=
4 −3
−2 1
T
=
4 −2
−3 1
5 A−1
= 1
|A| (Av
)T
= 1
−2
4 −2
−3 1
=

Ìàòðèö
Óðâóó
ìàòðèö
Æèøýý 10:
A =
1 2
3 4
1 |A| =
1 2
3 4
= 1 · 4 − 2 · 3 = 4 − 6 = −2 = 0 òóë óðâóó
2 A11 = (−1)1+1
· 4 = 4, A12 = (−1)1+2
· 3 = −3
A21
= (−1)2+1
· 2 = −2, A22 = (−1)2+2
· 1 = 1
3 Av
=
4 −3
−2 1
⇒
4 (Av
)T
=
4 −3
−2 1
T
=
4 −2
−3 1
5 A−1
= 1
|A| (Av
)T
= 1
−2
4 −2
−3 1
=
−2 1
1.5 −0.5

Ìàòðèö
Óðâóó
ìàòðèö
ÌàòðèöÓðâóó ìàòðèöûã îëîõ (íýãæ ìàòðèö àøèãëàõ àðãà)

Матриц болон тодорхойлогч, тэдгээрийг бодох нь

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