Дээд тоо - Лекц 7

Äýýä ýðýìáèéí òîäîðõîéëîã÷, Ìàòðèö, ò³³í äýýðõ ³éëäë³³ä

Äýýä òîî

Ëåêö 7

Ä.Áàòñóóðü

Õ³ì³³íëýãèéí Óõààíû Èõ Ñóðãóóëü

Áèçíåñèéí ñóðãóóëü

ÝÇÎ òýíõèì

Ä.Áàòñóóðü Äýýä òîî Ëåêö 7

Äýýä ýðýìáèéí òîäîðõîéëîã÷, Ìàòðèö, ò³³í äýýðõ ³éëäë³³ä

Àãóóëãà

1 Äýýä ýðýìáèéí òîäîðõîéëîã÷, Ìàòðèö, ò³³í äýýðõ ³éëäë³³ä

Äýýä ýðýìáèéí òîäîðõîéëîã÷

Ìàòðèö äýýðõ ³éëäë³³ä


Äýýä ýðýìáèéí òîäîðõîéëîã÷, Ìàòðèö, ò³³í äýýðõ ³éëäë³³äÄýýä ýðýìáèéí òîäîðõîéëîã÷Ìàòðèö äýýðõ ³éëäë³³ä


II , III -ð ýðýìáèéí òîäîðõîéëîã÷èéí á³òöèéã àæèãëàõàä ò³³íòýé

ò°ñòýéãýýð äóðûí n ýðýìáèéí òîäîðõîéëîã÷èéã òîäîðõîéëæ,

ò³³íèé òóñëàìæòàéãààð îëîí ³ë ìýäýãäýõòýé øóãàìàí

òýãøèòãýëèéí ñèñòåìèéã áîäîæ áîëîõ íü àæèãëàãäàâ. n2 øèðõýãýëåìåíòýýñ òîãòñîí n øèðõýã ì°ð, áàãàíàòàé äàðààõü êâàäðàò

òàáëèöûã àâ÷ ³çüå.

a11 a12 . . . a1na21 a22 . . . a2n. . . . . . . . . . . .an1 an2 . . . ann

aij -èéí ýõíèé èíäåêñ íü ì°ðèéí, õî¼ðäàõü íü áàãàíû äóãààðûã

èëýðõèéëíý.




Òîäîðõîéëîëò

Ýíýõ³³ êâàäðàò òàáëèöûí àëü íýã ì°ð (áàãàíà) -èéí á³õ

ýëåìåíò³³äèéí õàðãàëçàõ àëãåáðûí ã³éöýýëò (III -ð ýðýìáèéí

òîäîðõîéëîã÷èä òîäîðõîéëñíû àäèëààð òîäîðõîéëíî) -ýýð

³ðæ³³ëæ íýìñýí íèéëáýðèéã n ýðýìáèéí òîäîðõîéëîã÷ ãýæ

íýðëýýä∣∣∣∣∣∣∣∣a11 a12 . . . a1na21 a22 . . . a2n. . . . . . . . . . . .an1 an2 . . . ann

∣∣∣∣∣∣∣∣ = ai1Ai1 + ai2Ai2 + . . .+ ainAin

ãýæ òýìäýãëýíý.

III ýðýìáèéí òîäîðõîéëîã÷èéí á³õ ÷àíàðóóä n ýðýìáèéí

òîäîðõîéëîã÷èéí õóâüä õ³÷èíòýé.




n ýðýìáèéí òîäîðõîéëîã÷èéí àëü íýã ýëåìåíòèéí àëãåáðûí

ã³éöýýëò íü n − 1 ýðýìáèéí òîäîðõîéëîã÷ áàéõ á°ã°°ä ò³³íèéã

ì°í ì°ð áàãàíààð íü çàäëàõ çàìààð àëãåáðûí ã³éöýýëò íü 2, 3

-ð ýðýìáèéí òîäîðõîéëîã÷ ãàðàõ õ³ðòýë ì°ð áàãàíààð çàäëàõ

ïðîöåññûã ³ðãýëæë³³ëíý. Õàðèí äýýä ýðýìáèéí

òîäîðõîéëîã÷èéã áîäîõäîî òîäîðõîéëîã÷èéí ÷àíàðûã àøèãëàí

àëü íýã ì°ð áóþó áàãàíû íýãýýñ áóñàä ýëåìåíòèéã òýã áîëãîí

òýðõ³³ ì°ð áàãàíààð çàäëàæ áîäîõîä õÿëáàð áàéäàã.




Æèøýý∣∣∣∣∣∣∣∣∣∣10 2 1 −14 45 1 2 3 4

21 5 4 2 316 4 5 −9 511 3 4 5 1

∣∣∣∣∣∣∣∣∣∣òîäîðõîéëîã÷èéã áîä.

Áîäîëò: 2-ð áàãàíûã (-5) -ààð ³ðæ³³ëæ 1-ð áàãàíà äýýð íýìáýë∣∣∣∣∣∣∣∣∣∣0 2 1 −14 40 1 2 3 4−4 5 4 2 3−4 4 5 −9 5−4 3 4 5 1

∣∣∣∣∣∣∣∣∣∣áîëíî. Îäîî 3, 4-ð ì°ðèéã õàñâàë




Æèøýý∣∣∣∣∣∣∣∣∣∣0 2 1 −14 40 1 2 3 40 2 0 −3 20 1 1 −14 4−4 3 4 5 1

∣∣∣∣∣∣∣∣∣∣áîëîõ áà ³³íèéã 1-ð áàãàíààð çàäàëáàë

∣∣∣∣∣∣∣∣∣∣0 2 1 −14 40 1 2 3 40 2 0 −3 20 1 1 −14 4−4 3 4 5 1

∣∣∣∣∣∣∣∣∣∣= −4

∣∣∣∣∣∣∣∣2 1 −14 41 2 3 42 0 −3 21 1 −14 4

∣∣∣∣∣∣∣∣áîëíî.




Æèøýý

Èéìä áèä V ýðýìáèéí òîäîðõîéëîã÷èéã IV ýðýìáèéí

òîäîðõîéëîã÷èä øèëæ³³ëýâ. Ñ³³ëèéí òîäîðõîéëîã÷èéí 1-ð

ì°ð°°ñ 4-ð ì°ð°°ñ õàñâàë

−4

∣∣∣∣∣∣∣∣1 0 0 01 2 3 42 0 −3 21 1 −14 4

∣∣∣∣∣∣∣∣áîëíî. �³íèéã 1-ð ì°ð°°ð çàäàëáàë

−4

∣∣∣∣∣∣2 3 40 −3 21 −14 4

∣∣∣∣∣∣ = −200

áîëíî.Ä.Áàòñóóðü Äýýä òîî Ëåêö 7


Øóãàìàí òýãøèòãýëèéí ñèñòåì

a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2. . . . . . . . .an1x1 + an2x2 + . . .+ annxn = bn

n øèðõýã ³ë ìýäýãäýõòýé n øóãàìàí òýãøèòãýëèéí ñèñòåìèéã

àâ÷ ³çüå. Ýíý ñèñòåìèéí ³ë ìýäýãäýõ³³äèéí óðüäàõü

êîýôôèöèåíò³³äîîñ çîõèîñîí

∆ =

∣∣∣∣∣∣∣∣a11 a12 . . . a1na21 a22 . . . a2n. . . . . . . . . . . .an1 an2 . . . ann

∣∣∣∣∣∣∣∣òîäîðõîéëîã÷èéã ñèñòåìèéí òîäîðõîéëîã÷ ãýíý.




Õýðýâ ∆ 6= 0 áàéâàë °ã°ãäñ°í ñèñòåìèéí øèéä

x1 =∆1

∆, x2 =

∆2

∆, . . . xn =

∆n

∆

òîìú¼îãîîð èëýðõèéëýãäýíý. Ýíý íü °ìí° 2, 3 ³ë ìýäýãäýõòýé

àâ÷ ³çýæ áàéñàí Êðàìåðèéí ä³ðìèéí °ðã°òã°ë þì. Ýíä

∆i (i = 1, . . . , n) íü ñèñòåìèéí òîäîðõîéëîã÷èéí xi ³ëìýäýãäýõèéí óðüäàõü êîýôôèèåíòîîñ çîõèîñîí áàãàíûã ñóë

ãèø³³íýýð çîõèîñîí áàãàíààð ñîëèõîä ãàðàõ n ýðýìáèéí

òîäîðõîéëîã÷ þì.

Æèøýýx1 + 2x2 + 3x3 − 4x4 = 112x1 + x2 + 5x3 + x4 = 33x1 + 2x2 + x3 + 2x4 = −1x1 + x2 + 5x3 + x4 = 5

ñèñòåìèéã áîä.




Æèøýý

Áîäîëò: Ñèñòåìèéí òîäîðõîéëîã÷èéã áîäâîë

∆ =

∣∣∣∣∣∣∣∣1 2 3 −42 1 5 13 2 1 21 1 5 1

∣∣∣∣∣∣∣∣ = 54 6= 0 Èéìä °ã°ãäñ°í ñèñòåì Êðàìåðèéí

ä³ðìýýð áîäîãäîíî.

∆1 =

∣∣∣∣∣∣∣∣11 2 3 −43 1 5 1−1 2 1 25 1 5 1

∣∣∣∣∣∣∣∣ = −108 ∆2 =

∣∣∣∣∣∣∣∣1 11 3 −42 3 5 13 −1 1 21 5 5 1

∣∣∣∣∣∣∣∣ = 162




Æèøýý

∆3 =

∣∣∣∣∣∣∣∣1 2 11 −42 1 3 13 2 −1 21 1 5 1

∣∣∣∣∣∣∣∣ = 54 ∆4 =

∣∣∣∣∣∣∣∣1 2 3 112 1 5 33 2 1 −11 1 5 5

∣∣∣∣∣∣∣∣ = 54

áîëîõ òóë

x1 =−108

54= −2, x2 =

162

54, x3 =

54

54= 1, x4 =

54

54= 1

áàéíà.



Ìàòðèöûí òóõàé óõàãäàõóóí

Æèøýý

A3×2 =

2 3−1 04 2

íü 3× 2 õýìæýýñò òýãø °íö°ãò ìàòðèö

áîëíî. �ã°ãäñ°í æèøýýíèé õóâüä

a11 = 2, a12 = 3, a21 = −1, a22 = 0, a31 = 4, a32 = 2 áàéíà.


Õýðýâ ìàòðèöûí ì°ð áà áàãàíû òîî òýíö³³, °.õ m = n áàéâàë

ò³³íèéã n × n õýìæýýñò n ýðýìáèéí êâàäðàò ìàòðèö ãýíý.

Æèøýý

B3×3 =

−1 0 42 −5 3

12 4 5

ãóðàâäóãààð ýðýìáèéí êâàäðàò ìàòðèö.





a Êâàäðàò ìàòðèöûí ì°ð áîëîí áàãàíû íü äóãààð òýíö³³

áàéõ ýëåìåíò³³äèéã àãóóëñàí äèàãîíàëèéã ìàòðèöûí ãîë

äèàãîíàëü ãýíý. °.õ a11, a22, . . . , ann

á n ýðýìáèéí êâàäðàò ìàòðèöûí ãîë äèàãîíàëèéí

ýëåìåíò³³ä íü íýãòýé òýíö³³ áóñàä á³õ ýëåìåíò³³ä íü

òýãòýé òýíö³³ áîë °.õ{aij = 1, i = j ; i , j = 1, . . . naij = 0, i 6= j ; i , j = 1, . . . n

áîë ³³íèéã íýãæ ìàòðèö ãýýä En×n ãýæ òýìäýãëýäýã. Çàðèì

òîõèîëäîëä In×n ãýæ ÷ òýìäýãëýäýã.




Íýãæ ìàòðèö íü øóãàìàí àëãåáðò ÷óõàë ³³ðýã ã³éöýòãýäýã áà

àðèôìåòèê òîîëëûí 1 íýãæ ýëåìåíòòýé ò°ñòýé ÷àíàðòàé áàéäàã.

Æèøýý

Òîäîðõîéëîëò ¼ñîîð 2-ð ýðýìáèéí íýãæ ìàòðèö íü

E2×2 =

(1 00 1

)3-ð ýðýìáèéí íýãæ ìàòðèö

E3×3 =

1 0 00 1 00 0 1

áàéíà.




Õî¼ð ìàòðèöûí ì°ð áà áàãàíû òîîíóóä õàðãàëçàí òýíö³³ áîë

èæèë õýìæýýñò ìàòðèöóóä ãýíý. Èæèë õýìæýýñò ìàòðèöóóäûí

õàðãàëçàõ áàéðëàë äàõü ýëåìåíò³³ä òýíö³³ áîë òýäãýýðèéã

òýíö³³ ìàòðèöóóä ãýíý.

Æèøýý

A3×2 =

1 2−1 00 −2

; B3×2 =

4 −30 −1−2 0

ìàòðèöóóä èæèë õýìæýýñò ìàòðèöóóä áîëîâ÷ òýíö³³ áèø áàéíà.




Æèøýý

A2×2 =

(2x −15 4y − 5

); B2×2 =

(8− 3y −1

5 3x

)ìàòðèöóóä òýíö³³ áàéõ x , y -èéí óòãûã îë.

Áîäîëò: Ìàòðèöóóä òýíö³³ áàéõûí òóëä èæèë õýìæýýñòýé

áàéõààñ ãàäíà õàðãàëçàõ ýëåìåíò³³ä òýíö³³ áàéõ ¼ñòîé.

Òèéìýýñ òýíö³³ áàéõ í°õöë°°ñ äàðààõ òýãøèòãýëèéí ñèñòåìèéã

çîõèîæ áîëíî.{2x = 8− 3y4y − 5 = 3x

⇒{

2x + 3y = 8−3x + 4y = 5

⇒{

x = 8−3y2

−3(8−3y

2

)+ 4y = 5

Ýíýõ³³ òýãøèòãýëèéí ñèñòåìèéí øèéä íü x = 1, y = 2 ãýæ

ãàðíà. Ýíý í°õö°ëä õî¼ð ìàòðèö òýíö³³ áàéíà.





Ãîë äèàãîíàëü áîëîí ò³³íèéõýý ç°âõ°í íýã òàëä òýãýýñ ÿëãààòàé

ýëåìåíòòýé, áóñàä áàéðëàëäàà äàí òýã ýëåìåíòòýé êâàäðàò

ìàòðèöûã ãóðâàëæèí ìàòðèö ãýíý. Õýðâýý ãîë äèàãîíàëèéíõàà

äýýä òàëä òýãýýñ ÿëãààòàé ýëåìåíò³³äòýé áîë äýýðýý

ãóðâàëæèí, õàðèí ãîë äèàãîíàëèéíõàà äîîä òàëä òýãýýñ

ÿëãààòàé ýëåìåíò³³äòýé áîë äîîðîî ãóðâàëæèí ìàòðèö ãýíý

Æèøýý

A4×4 =

1 8 7 10 2 0 50 0 −1 30 0 0 4

; B4×4 =

1 0 0 04 −1 0 0−5 6 0 04 0 7 −2

A íü äýýðýý ãóðâàëæèí, B íü äîîðîî ãóðâàëæèí ìàòðèö.





à Ç°âõ°í ãîë äèàãîíàëü äýýðýý òýãýýñ ÿëãààòàé, áóñàä

áàéðëàëäàà òýã ýëåìåíòòýé êâàäðàò ìàòðèöûã äèàãîíàëü

ìàòðèö ãýíý. Äèàãîíàëü ìàòðèöûã èõýâ÷ëýí Dn×n ³ñãýýð

òýìäýãëýäýã. Íýãæ ìàòðèö íü äèàãîíàëü ìàòðèöûí íýã

òóõàéí òîõèîëäîë þì.

á Á³õ ýëåìåíò íü òýã áàéõ ìàòðèöûã òýã ìàòðèö ãýýä 0n×n

ãýæ òýìäýãëýå.

â Õýðâýý äèàãîíàëü ìàòðèöûí ãîë äèàãîíàëü äýýðõ á³õ

ýëåìåíò íü òýíö³³ áàéâàë ò³³íèéã ñêàëÿð ìàòðèö ãýíý.



Ìàòðèö äýýðõ ³éëäë³³ä

Ìàòðèö äýýð äàðààõ ³éëäë³³äèéã òîäîðõîéëæ áîëíî.

1 Ìàòðèöóóäûã íýìýõ

2 Ìàòðèöûã òîîãîîð ³ðæ³³ëýõ

3 Ìàòðèöóóäûã õàñàõ

4 Ìàòðèöûã ìàòðèöààð ³ðæ³³ëýõ



Ìàòðèöóóäûã íýìýõ

Èæèë õýìæýýñò Am×n = aij , i = 1, 2, . . . ,m; j = 1, 2, . . . , n áà

Bm×n = bij , i = 1, 2, . . . ,m; j = 1, 2, . . . , n ìàòðèöóóäûí íèéëáýð

ãýæ õàðãàëçàõ áàéðëàë äàõü òîîíóóäûã íýìýõýä ³³ñýõ ìàòðèöûã

õýëýõ á°ã°°ä ò³³íèéã Cm×n = cij , i = 1, 2, . . . ,m; j = 1, 2, . . . , nãýæ òýìäýãëýâýë cij = aij + bij , i = 1, 2, . . .m; j = 1, 2, . . . , náîëíî. Õî¼ð ìàòðèöûã íýìýõ ³éëäëèéí õóâüä òàâèãäàõ ³íäñýí

øààðäëàãà íü òóõàéí õî¼ð ìàòðèö èæèë õýìæýýñòýé áàéõ ÿâäàë

þì. Ì°í íèéëáýð ìàòðèö íü èæèë õýìæýýñò ìàòðèö ãàðàõ íü

îéëãîìæòîé.




Cm×n = Am×n + Bm×n =a11 a12 . . . a1na21 a22 . . . a2n...

.... . .

...

am1 am2 . . . amn

+

b11 b12 . . . b1nb21 b22 . . . b2n...

.... . .

...

bm1 bm2 . . . bmn

=

a11 + b11 a12 + b12 . . . a1n + b1na21 + b21 a22 + b22 . . . a2n + b2n

......

. . ....

am1 + bm1 am2 + bm2 . . . amn + bmn




Æèøýý

A2×2 =

(1 23 4

), B2×2 =

(5 67 8

)ãýæ °ã°ãäñ°í áîë

C2×2 = A2×2 + B2×2 ìàòðèöûã îë. Áîäîëò: Òîäîðõîéëîëò ¼ñîîð

C2×2 = A2×2 + B2×2 =

(1 23 4

)+

(5 67 8

)=

(1 + 5 2 + 63 + 7 4 + 8

)=(

6 810 12

)áîëíî.

Ìàòðèöóóäûã íýìýõ ³éëäëèéí õóâüä

1 A + B = B + A áàéð ñýëãýõ õóóëü

2 (A + B) + C = A + (B + C ) á³ëýãëýõ õóóëü

3 A + 0 = A òýã ìàòðèöûã íýìýõ õóóëü

÷àíàðóóä áèåëäýã.



Ìàòðèöóóäûã òîîãîîð ³ðæ³³ëýõ

Ìàòðèöûã òîîãîîð ³ðæ³³ëýõ ³éëäýë ãýæ ò³³íèé á³õ ýëåìåíòèéã

°ãñ°í òîîãîîð ³ðæ³³ëýõèéã õýëíý. �.õ °ãñ°í

Am×n, i = 1, 2, . . . ,m; j = 1, 2, . . . n ìàòðèö áîëîí λ 6= 0 òîîíû

õóâüä λ · Am×n = (λ · aij), i = 1, 2, . . .m; j = 1, 2, . . . n ãýæ îëíî.

Òîîãîîð ³ðæ³³ëýõ ³éëäëèéí õóâüä ìàòðèöàä ÿìàð íýãýí

øààðäëàãà òàâèõã³é.

λ·Am×n = λ·

a11 a12 . . . a1na21 a22 . . . a2n...

.... . .

...

am1 am2 . . . amn

=

λa11 λa12 . . . λa1nλa21 λa22 . . . λa2n...

.... . .

...

λam1 λam2 . . . λamn

áîëíî.



Ìàòðèöûã òîîãîîð ³ðæ³³ëýõ

Æèøýý

1 A2×2 =

(−1 25 −4

)ìàòðèö °ã°ãäñ°í áîë 2A-ã îë.

Áîäîëò:

2A = 2 ·(−1 25 −4

)=

(2 · (−1) 2 · 2

2 · 5 2 · (−4)

)=

(−2 410 −8

)2 A =

(1 23 4

), B =

(5 67 8

)ìàòðèöóóä °ã°ãäñ°í áîë

3A + 2B -ã îë.

Áîäîëò: 3A + 2B = 3

(1 23 4

)+ 2

(5 67 8

)=(

3 69 12

)+

(10 1214 16

)=

(3 + 10 6 + 129 + 14 12 + 16

)=

(13 1823 28

)áîëíî.



Ìàòðèöûã òîîãîîð ³ðæ³³ëýõ

Ìàòðèöûã òîîãîîð ³ðæ³³ëýõ áîëîí íýìýõ ³éëäëèéí õóâüä

äàðààõ ÷àíàðóóä áèåëíý.

1 λ · A = A · λ áàéð ñýëãýõ

2 (λ+ µ) · A = λ · A + µ · A õààëò çàäëàõ

3 λ · (A + B) = λ · A + λ · B õààëò çàäëàõ

4 1 · A = A íýãæýýð ³ðæ³³ëýõ

5 0 · A = 0 òýãýýð ³ðæ³³ëýõ



Ìàòðèöóóäûã õàñàõ

Ìàòðèöóóäûí íýìýõ áà ò³³íèéã òîîãîîð ³ðæ³³ëýõ ³éëäë³³äèéí

òóñëàìæòàé ìàòðèöóóäûã õàñàõ ³éëäëèéã òîäîðõîéëæ áîëíî.

�.õ A ìàòðèöààñ B ìàòðèöûã õàñàõ ³éëäëèéã A ìàòðèö äýýð B

ìàòðèöûã (−1) -ýýð ³ðæ³³ëæ íýìýõ áàéäëààð òîäîðõîéëîãäîíî.

A− B = A + (−1)B

áîëíî.

Æèøýý

A =

(53

), B =

(42

)áîë A− B -ã îë.

Áîäîëò: Òîäîðõîéëîëò ¼ñîîð A− B = A + (−1)B =(53

)+ (−1) ·

(42

)=

(53

)+

(−4−2

)=

(5 + (−4)3 + (−2)

)=

(11

)áîëíî.



Ìàòðèöóóäûã ³ðæ³³ëýõ

Am×n = aij , i = 1, 2, . . . ,m; j = 1, 2, . . . , n; Bp×q = bij , i =1, 2, . . . , p; j = 1, 2, . . . , q õî¼ð ìàòðèö °ã°ãäñ°í áîëîã. Õýðâýý

³ðæâýðèéí ýõíèé ìàòðèöûí áàãàíûí òîî õî¼ð äàõü ìàòðèöûí

ì°ðèéí òîîòîé òýíö³³ áàéâàë õî¼ð ìàòðèöûã ³ðæ³³ëæ áîëíî.

Ýíýõ³³ ÷àíàð áèåëæ áàéãàà õî¼ð ìàòðèöóóäûã íèéöòýé

ìàòðèöóóä ãýíý. �.õ ìàòðèöóóäûí ³ðæâýð çààâàë

òîäîðõîéëîãäîõ àëáàã³é á°ã°°ä AB ³ðæâýð òîäîðõîéëîãäñîí

áàéëàà ÷ BA òîäîðõîéëîãäîõã³é áàéæ áîëîõ áà, BAòîäîðõîéëîãäñîí áàéëàà ÷ AB -òýé òýíö³³ áèø ýñâýë AB -ýýñ

õýìæýýñ °°ð áàéæ áîëíî. �³íèé ó÷èð íü ìàòðèöóóäûã ³ðæ³³ëýõ

³éëäëèéí øààðäëàãàòàé õîëáîîòîé. Õýðýâ A áà B íü èæèë

ýðýìáèéí êâàäðàò ìàòðèöóóä áîë AB,BA ³ðæâýð³³ä ³ðãýëæ

áèåëýõ òîäîðõîéëîãäîõ áà òýäãýýðèéí õóâüä AB = BA ÷àíàð

áèåëýõ àëáàã³é áàéäàã.




Am×n = aij , i = 1, 2, . . . ,m; j = 1, 2, . . . , n; Bp×q = bij , i =1, 2, . . . , p; j = 1, 2, . . . , q ìàòðèöóóäûí õóâüä n = p ³åä äýýðõ

õî¼ð ìàòðèöûí ³ðæâýð òîäîðõîéëîãäîíî. Ýíý í°õö°ëä AB³ðæâýðèéã C ãýæ òýìäýãëýâýë Cm×q = Am×nBn×q áîëîõ áà Cìàòðèöûí ýëåìåíò cij , i = 1, 2, . . . ,m; j = 1, 2, . . . q íü

cij =(ai1 ai2 . . . ain

)b1jb2j...

bnj

=

= ai1 · b1j + ai2 · b2j + . . . ain · bnj =n∑

k=1

aik · bkj




�.õ ³ðæâýð ìàòðèöûí cij ýëåìåíòèéã îëîõäîî ýõíèé³ðæèãäýõ³³í ìàòðèöûí i -ð ì°ðèéí ýëåìåíò³³äèéã õî¼ð äàõü

³ðæèãäýõ³³í ìàòðèöûí j -ð áàãàíû ýëåìåíò³³äýýð õàðãàëçóóëàí

³ðæ³³ëæ, òýäãýýðèéã íýìæ îëäîã.

Æèøýý

A3×3 =

a11 a12 a13a21 a22 a23a31 a32 a33

, B3×2 =

b11 b12b21 b22b31 b32

ìàòðèöóóäûí ³ðæâýðèéã òîäîðõîéëæ, ³ðæâýð ìàòðèöûí

ýëåìåíò³³äèéã îë.

Áîäîëò: Áîäëîãîä °ã°ãäñí°°ð m = 3, n = p = 3; q = 2áàéãàà òóë ýíýõ³³ õî¼ð ìàòðèöûã ³ðæ³³ëæ áîëíî. Òýãâýë A · B³ðæâýðýýð ³³ñýõ C ìàòðèöûí ýëåìåíò³³äèéã ³ðæâýðèéí

ä³ðìýýð òîäîðõîéëú¼. �ðæâýð ìàòðèöûí õýìæýýñ íü 3× 2 áàéõ

áîëíî.Ä.Áàòñóóðü Äýýä òîî Ëåêö 7



Æèøýý

C3×2 = A3×3·B3×2 =

c11 c12c21 c22c31 c32

=

a11 a12 a13a21 a22 a23a31 a32 a33

·b11 b12b21 b22b31 b32

�ðæâýð ìàòðèöûí ýëåìåíò³³äèéã íýã á³ð÷ëýí áè÷âýë:

c11 = a11b11 + a12b21 + a13b31; c12 = a11b12 + a12b22 + a13b32;

c21 = a21b11 + a22b21 + a23b31; c22 = a21b12 + a22b22 + a23b32;

c31 = a31b11 + a32b21 + a33b31; c32 = a31b12 + a32b22 + a33b32;

ãýæ îëäîíî.




Æèøýý

A2×2 =

(1 23 4

), B2×2 =

(5 67 8

)ìàòðèöóóä °ã°ãäñ°í áîë AB,BA -ã îë.

Áîäîëò:

A2×2 · B2×2 =

(1 23 4

)·(

5 67 8

)=

(1 · 5 + 2 · 7 1 · 6 + 2 · 83 · 5 + 4 · 7 3 · 6 + 4 · 8

)=

(19 2243 50

)B2×2 · A2×2 =

(5 67 8

)·(

1 23 4

)=

(5 · 1 + 6 · 3 5 · 2 + 6 · 47 · 1 + 8 · 3 7 · 2 + 8 · 4

)=(

23 3431 46

)áîëíî.





n ýðýìáèéí êâàäðàò ìàòðèöûí õóâüä A · A−1 = A−1 · A = Eí°õöëèéã õàíãàæ áóé A−1 ìàòðèöûã °ãñ°í A ìàòðèöûí óðâóó

ìàòðèö ãýíý.

Æèøýý

A2×2 =

(1 21 3

)ìàòðèöûí õóâüä A−1 =

(3 −2−1 1

)ìàòðèö

óðâóó ìàòðèö íü áîëîõûã øàëãà.

Áîäîëò:

A · A−1 =

(1 21 3

)·(

3 −2−1 1

)=

=

(1 · 3 + 2 · (−1) 1 · (−2) + 2 · 11 · 3 + 3 · (−1) 1 · (−2) + 3 · 1

)=

(1 00 1

)= E




Æèøýý

A−1 · A =

(3 −2−1 1

)·(

1 21 3

)=

=

(3 · 1 + (−2) · 1 3 · 2 + (−2) · 3(−1) · 1 + 1 · 1 (−1) · 2 + 1 · 3

)=

(1 00 1

)= E

Ìàòðèöóóäûã ³ðæ³³ëýõ ³éëäëèéí õóâüä áîäèò òîîíû ³ðæâýðèéí

õóâüä íýã óòãàòàé áèåëäýã çàðèì ÷àíàðóóä ç°ð÷èãää°ã.

Òóõàéëáàë äóðûí 2 áîäèò òîîíû ³ðæâýð òýã áîë òýäãýýð

òîîíóóäûí äîð õàÿæ íýã íü òýã áàéäàã õàðèí ìàòðèöûí

³ðæâýðèéí ä³ðìýíä ýíý ä³ðýì áèåëýõã³é áàéæ áîëäîã. �³íèéã

äàðààõ æèøýýãýýð õàðóóëúÿ.




Æèøýý

A2×2 =

(1 11 1

), B2×2 =

(−1 −11 1

)ìàòðèöóóäûí õóâüä äýýð

äóðüäñàí ÷àíàðûã øàëãà.

Áîäîëò: A · B =

(1 11 1

)·(−1 −11 1

)=

(−1 + 1 −1 + 1−1 + 1 −1 + 1

)=(

0 00 0

)= O2×2

Ìàòðèöûã ³ðæ³³ëýõ ³éëäëèéí õóâüä äàðààõ ÷àíàðóóä áèåëäýã.

1 A · (B + C ) = A · B + A · C õààëò çàäëàõ

2 (A + B) · C = A · C + B · C õààëò çàäëàõ

3 A · B · C = (A · B) · C = A · (B · C ) á³ëýãëýõ


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