MATRICE seminarski

UNIVERZITET U SARAJEVU FARMACEUTSK FARMACEUTSKII FAKULT FAKULT

Predmet: Matematika

MATRICE

Mentor: prof.dr. Senada Kala!"#$

St!dent: Am#na Mar#$ %ro& #nde'(a:))*+,-

Sara&e/o0 de1emar 23-.

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Sadržaj:

-.3 U/od ............................................. .................................................................... .............................................. .............................................. ..........................................4 ...................4 -.- 5ef#n#1#&a # o(no/na (/o&(t/a matr#1a ............................................ ................................................................... ...................................... ............... -.2 6pera1#&e (a matr#1ama ............................................ ................................................................... .............................................. .....................................) ..............) -.2.- Sa#ran&e # od!7#man&e matr#1a ............................................. ..................................................................... ............................................) ....................) -.2.2 Mno8en&e matr#1a ('alarom ............................................ ................................................................... ............................................) .....................) -.2.4 Mno8en&e matr#1a matr#1om ............................................ ................................................................... ...........................................9 ....................9 -.4 Tran(pono/ane Tran(pono/ane matr#1e ............................................ ................................................................... .............................................. .................................... ............. -. K/adratne matr#1e ............................................. .................................................................... ............................................... ........................................... ................... -..-S#metr#;ne # 'o(o(#metr#;ne matr#1e ............................................ ................................................................... ..................................... .............. -..2 Jed#n#;ne matr#1e ............................................. .................................................................... ............................................... .....................................* .............* -..4 In/er7ne '/adratne matr#1e .............................................. ..................................................................... ...........................................* ....................* -.. Re






1.0 Uvod

Jo" ! IV (tol&e$! pr#&e no/e ere %a#lon1# (! (e 'or#(t#l# taelama # n&#=o/#m 'ara'ter#(t#'ama 'o&e l#;e na ono "to dana( na7#/amo matr#1ama. Me>!t#m0 pradomo/#nom matr#1a (matra (e K#na. U II (tol&e$! pr#&e no/e ere ?an 5@na(t@ &e pred(ta/#o r&e"en&e prolema pr#no(a r#8e# to !pra/o matr#;nom metodom. To &e #o 'la(#;an na;#n r&e"a/an&a (#(tema l#nearn#= &edna;#na al# (e 7o< 7ap#(a do/od# ! /e7! (a matr#1ama. Matr#1e po(ta&! pra/# d#o matemat#'e matemat#'e te' ! I (tol&e$!. Cantor Cantor # Lee< (! teor#&om '/anta '/anta do"l# do matr#1a0 'o&e ! matemat#'! !/od# Art=!r Art=!r Ca@le@ -*).






1.1 Definicija i osnovna svojstva matrica

Matr#1e &e ! matemat#'! !/eo en
pre(l#'a/a 5e'arto/ 5e'arto/ pro#7/od -020...0m-020....0n -020...0m-020....0n. .

Matr#1a t#pa mxn &e pra/o!< pra/o!
A'o &e A matr#1a matr#1a formata mxn0 tada n&ene elemente o7na;a/amo (a a #& 0
•

•

Matr#1a 'o&a (adr8# (amo &edn! /r(t! 7o/e (e matr#1a /r(te #l# /e'torG/r(ta0 a a'o (adr8#(amo &edn! 'olon! 7o/e (e matr#1a 'olona #l# /e'torG'olona. /e'torG'olona. A'o (! element# matr#1e realn# ro&e/# tada &e matr#1a realna0 a !'ol#'o (adr8# 'omple'(ne elemente matr#1a &e 'omple'(na. 'omple'(na. Matr#1a ;#&# (! (/# element# n!le na7#/a (e n!la matr#1a.

Jedna'o(t matr#1a Za d/#&e matr#1e A  a #&Hmn # %  #&Hmn'a8emo da (! &edn'a onda # (amo onda a'o (! #(to< t#pa # a'o (! #m od
[

2

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j + 3

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6/e matr#1e (! #(to< t#pa mxn 24 pa &e #(p!n&en potrean !(lo/ 7a &edna'o(t matr#1a. Trea Trea &o" #7&edna;#t# od
j G-

73

1.2 !eracije sa matricama

-.2.-Sa#ran&e # od!7#man&e matr#1a Definicija 2: 5/#&e matr#1e #(to< t#pa A  a #&Hmn #

od
%  #&Hmn (a#ra&! (e ta'o "to #m (e (aer!

A Q %  a#&Hm  n Q #&Hm  n  a#& Q #&Hm  n Za (a#ran&e matr#1a /a8e 7a'on#: -. Kom!ta1#&e: A Q %  % Q A 2. A(o1#&a1#&e: A(o1#&a1#&e: A Q % Q CH  A Q %H Q C 4. Ne!traln# element 7a (a#ran&e matr#1a t#pa mxn &e n!la matr#1a t#pa mxn. Za (/a'! matr#1! A  a #&H po(to&# (!protna matr#1a #(to< t#pa A  a #&H ta'o da &e: AQAH  a#& HQa#&H  a#&a#&H  o#&H  6 #7 ;e
(

) (

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−1

Q

−2

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11

3

) ( 

1− 2

0+ 3

1 +2

−2 + 1

3+ 4

4 −2

7 +7

− 12+ 11

−1 + 3

Definicija 3:5/#&e

)(

−1

3

3

= −1

7

2

−1

2

14

)

matr#1e #(to< t#pa A  a #&Hmn # %  #&Hmnod!7#ma&! (e ta'o "to #m (e od!7m! od
A%

(

1−

(−2 )

0−3

− 2− 1

3 −4

7 −7

− 12−11

1 −2 4−

−1− 3

-.2.2Mno8en&e -.2.2Mno8en&e matr#1a ('alarom

) (

(−2 ) 

3

−3

−1

−3

−1

6

0

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−4

)







A'o &e data matr#1a A  a #&Hmn# realan ro& 0 tada &e A  a #&Hmn Pr.4 5ata &e matr#1a A 

2A 

(

(

)

1

0

1

−2

3

4

7

− 12

−1

1· 2

0·2

1·2

−2 · 2

3 ·2

4·2

7 ·2

− 12 · 2

−1 · 2

) ( 

0 pomno8#t# matr#1! ('alarom a'o &e 2

2

0

2

−4

6

8

14

− 24

−2

)

Za mno8en&e matr#1a ('alarom /a8#: -. Za'on 'om!ta1#&e: A  A 2. Za'on a(o1#&a1#&e: HA  A  AH 4. 5/a 7a'ona d#(tr#!1#&e:  Q HA  A Q A

#

A Q %H  A Q %

. Za'on &ed#n#1e: -A  A-.2.4 Mno8en&e matr#1e matr#1om Definicija5:Matr#1e Matr#1e A 

a#&Hmn # %   #&H  p mo
∑ A  %  C1 #&Hm  p1#& 

k =-

aikb jk

Pr.4Naći Pr.4 Naći proizvod matrica

A

(

0

1

−1

0

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3

2

0

2

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2

)

#%

4

( ) 1

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Matr#1e (! (a








Pro#7/od d/#&e matr#1e mo8e #t# n!la matr#1a#a'o n#&edan fa'tor n#&e n!la.

(

Pr.)A Pr.) A%

) ( ) ( )

1

−1

1

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2

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1

0

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1

2

3

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6

1

2

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Za mno8en&e matr#1a matr#1ama /a8#: -. Za'on a(o1#&a1#&e: a(o1#&a1#& e: A%C  A %CH 2. Za'on d#(tr#!1#&e ( l#&e/a l#& e/a prema (a#ran&! A%QCH  A% Q AC 4. Za'on d#(tr#!1#&e ( de(na prema (a#ran&! %QCH A  %A Q CA . A3  3A  3 ). Mno8en&e matr#1a ! op"tem (l!;a&! n#&e 'om!tat#/no. A'o po(to&# pro#7/od matr#1a A # %0 odno(no A%0 to ne 7na;# da mora da po(to&# pro#7/od % # A0 odno(no %A. Wa' # a'o po(to&# A% ne mora #t# &edna'o %A. Me>!t#m0 a'o /a8# A%  %A onda (e 'a8e da (! matr#1e 'om!tat#/ne. 1." Trans!onovane matrice Definicija 6: Tran(pono/ana matr#1a matr#1e A t#pa m x n &e

matr#1a AT t#pa m x n 'o&a (e od matr#1e A do#&a ta'o "to /r(te matr#1e A 7am#&ene m&e(ta (a od
A

(

a a a

11 21 31

a

a

a

a

a

a

12 22 32

13 23 33

)

AT 

(

a a a

11 12

13

a

a

a

a

a

a

21 22 23

31 32 33

)

Za tran(pono/ane matr#1e /a8#: -. ATHT  A 2. AHT   AT 4. A Q %H T  AT Q %T A'o (! date matr#1e A  a #&Hmn # %a#&Hmn tada &e A%HT  %TAT 1.# $vadratne matrice







-..-S#metr#;ne # 'o(o(#metr#;ne matr#1e K/adratna matr#1a (e na7#/a simetri&na matrica matric a a'o (! n&en# element#0 'o&# le8e (#metr#;no ( o7#rom na !(ono &edna'# aija ji. S#metr#;na matr#1a &edna'a &e (/o&o& tran(pono/ano& matr#1# A T  A t&. A  X #l# antisimetri&na matricaA  X&e matr#1a 'od 'o&e (! element#0 (#metr#;no ra(pore>en# ! odno(! na
'(s(simetri&na matrica

Za (#metr#;ne # ant#(#metr#;ne matr#1e /r#&ed# teorem: S/a'a S/a 'a '/adr '/adratn atnaa matr#1 matr#1aa A(e A(e mo8e mo8e ra(ta/ ra(ta/#t# #t# ! 7#r 7#r &edne &edne (#metr (#metr#;n #;nee matr#1 matr#1ee A(# &edne &edne ant#(#metr#;ne matr#1e Aa0 t&.

-..2Jed#n#;ne matr#1e Jed#n#;na matr#1a &e '/adratna matr#1a 'od 'o&e (! (/# element# na d#&a
( ) 1

0

0

0

1

0

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44

Jed#n#;n! matr#1! o7na;a/amo (a *. 6/a matr#1a (e &o" na7#/a #dent#;nom &er ! pro#7/od! (a dr!<#m matr#1ama da&e !pra/o n&#= 'ao re7!ltat mno8en&a0 odno(no ne m#&en&a #=. AIIAA -..4In/er7ne '/adratne matr#1e Definicija +:

In/er7 In/er7na na matr#1 matr#1aa A-ne'e ne'e '/adra '/adratne tne matr#1 matr#1ee A def#n# def#n#(an (anaa &e (/o&(t/ (/o&(t/om om da







Za ne'! '/adratn! matr#1! A$emo re$# da &e (rt(g(nalna matrica a'o &e pro#7/od te matr#1e # n&o& tran(pono/ane matr#1e X &edna' &ed#n#;no& matr#1# I. Prod!'t matr#1e A # tran(pono/ane matr#1e X da&e (#metr#;n! matr#1! %: AX  % &er &e:

1.% Determinanta &vadratne matrice

5eterm#nanta '/adratne matr#1e A ∈Rnxn &e realan ro& pr#dr!8en to& matr#1# # o7na;a/amo &e H. (a  #l# det  Pr.9 aH 5ata &e matr#1a A 

(

4

−1

3

2

) 0 na$# n&en! determ#nant!:

5eterm#nanta dr!






-.)- Ra;!nan&e #n/er7ne matr#1e !7 !potre! determ#nante K/adratna matr#1a0 'a'o &e ran#&e re;eno0 &e re

A[  A#&HT

6dre>#/an&e #n/er7ne matr#1e: -. I7ra;!namo determ#nant! matr#1e A. U'ol#'o &e detAH Y 3 prela7#mo na naredn# 'ora' 2. 6dred#mo 'ofa'tore A #&(/#= elemenata a #& matr#1e A 4. Form#ramo matr#1! 'ofa'tora pa &e tran(pon!&emo # na ta& na;#n do#&amo ad&!n
(

1

2

0

−2

3

1

0

2

−1

detAH  G+ A--  G) A-2  G2 A-4  G A2- 2A22 G-A24  G2

(

−5

−2

−4

2

−1

−2

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7

)

)







( ) 5

1

A  G-

−9

(

−5

2

2

−2

−1

−1

−4

−2

7

)



− 2 −2

9

9

9

2

1

1

9

9

9

4

2

7

9

9

9

1.' (rimjena matrica na rje)avanje matri*ni+ jedna*ina i sistema ,inearni+ jedna*ina

6pera1#&e mno8en&a matr#1a # #n/er7ne matr#1e /e7ane (! 7a r&e"en&e (#(tema l#nearn#=









A  matr#1a (#(tema

  /e'tor 'olone

%  (loodn# ;lano/# (#(tema

A   % 5a # odred#l# odred#l# matr#1! matr#1! 0 matr#1a matr#1a A mora da !de ne(#nen0 a'o &e ('!p r&e"en&a r &e"en&a &edno;lan. U (!protnom &e neodre>en. A'o (! (/# (loodn# ;lano/# (#(tema n!le0 ta'a/ (#(tem (e na7#/a =omo






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