atps - matrizes e determinantes

7/26/2019 Atps - Matrizes e Determinantes

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CURSO DE ENGENHARIA MECNICA TURMA: 1SEM.

ANDERSON MACRIO RA 1587919464EWERTON FERNANDES DE BARROS RA 1593868112FELIPE GERALDO NOGUEIRA RA 1426151578FRANCISCO FERREIRA DA SILVA RA 2485709302GUSTAVO GIATTI RA 1423175962MRCIO ALAN BEERRA DA COSTA RA 1416237429MARCOS VIERA DOS SANTOS RA 1429119313MAT!EUS RODRIGUES DE SOUA RA 1436944181

MIC!EL ULISSES OLIVEIRA SANTOS RA 2484653851VIN"CIUS RODRIGUES DA COSTA RA 2485761190

LGEBRA LINEAR COM GEOMETRIAANALTICA E VETORES

ATIVIDADES PRTICAS SUPERVISIONADAS:

MATRIZES E DETERMINANTES

T#$%$&'( $)#*+*,-$.( $( C*,-#( U,/*#+/-#/(.* S$,-$ B#%$#$ DO*+-* (( )$#-* .*A-//.$.*+ P#-/$+ S)*#/+/(,$.$+ )$#$ $./+/)&/,$ .* &*%#$ L/,*$# .( #+( .*

E,*,'$#/$ M*,/$

O#/*,-$.(#:P#(; R(

7/26/2019 Atps - Matrizes e Determinantes

2/17

1 I,-#(.>?( 3

2 D*;/,/>?( .* $ M$-#/= 4

3 A)#*+*,-$>?( .* $ M$-#/= 4

4 T/)(+ .* M$-#/=*+5

41 M$-#/= L/,'$ 5

42 M$-#/= C(&,$ 5

43 M$-#/= @$.#$.$ 6

431 D/$(,$& P#/,/)$& 6

432 D/$(,$& S*,.$#/$ 6

44 M$-#/= N&$ ( *#( 7

45 M$-#/= I.*,-/.$.* ( U,/-#/$7

46 M$-#/= T#$,+)(+-$ 8

47 M$-#/= S/?( * S%-#$>?( .* M$-#/=*+10

52 M&-/)&/$>?( .* M$-#/=*+ 10

6 D*-*#/,$,-*+ 11

61 D*-*#/,$,-*+ 1 O#.*12

62 D*-*#/,$,-*+ 2 O#.* 12

63 D*-*#/,$,-*+ 3 O#.* 13

7 C(,&+?(

8 B/%&/(#$;/$

7/26/2019 Atps - Matrizes e Determinantes

3/17

3

1. Intr!"#$

A $-*-/$ )(++/ #/$+ #*#$+ * ;(#&$+ #/$.$ )$#$ ;$/&/-$# (+ &&(+ ,( ./$ ./$ .(+*# '$,( M/-$+ .$+ *=*+ ,$ /,/$ * ,$ $-*-/$ +?( (#$,/=$.$+ * &/,'$+ * (&,$+

;(#$,.( * (,,-( #*-$,&$#*+ '$$.(+ .* $-#/= P(.*(+ #*)#*+*,-$# $ $-#/=

)*&( +%(&( $/ (,.* /: &/,'$+ * : (&,$+ T(.$ $-#/= < ./+)(+-$ ,$ ;(#$ H , *# ./=*#

$ -$%*&$ .* &/,'$+ '(#/=(,-$/+ * , &/,'$+ *#-/$/+

A+ $-#/=*+ ,?( +* *,(,-#$ +(*,-* ,(+ *+-.(+ .$ $-*-/$ $+ -$%?( .* $-*#/$& ,$ (,+-#>?( .* $ *+-#-#$ .*

++-*,-$>?( .* $ &$*

F+/$ < ;*/-( ( +( .$+ $-#/=*+ $ )$#-/# .* -$%*&$+ #*&$/(,$,.( ( .*+&($*,-( * ( -*)(

S&*+-*# $/,.$ /$ $+ $-#/=*+ (( *#( /,#*./*,-* .(+ .*-*#/,$,-*+ * %&(( .*

#*-$,&$# .* -*#(+ * )$#$ *&* #*)#*+*,-$$ .*-*#/,$,-* * +* ;(++* ( $

$-#/= $ )$#-/# .( $& )(.*(+ ;$=*# /*,+(+ +/+-*$+ .*-*#/,$,-*+ S(*,-* ( C$&*

* *&$+ )$++$ $ -*# /.$ )#J)#/$ * #$.$-/$*,-* (*>$ $ +)&$,-$# (+ .*-*#/,$,-*+

* /)(#-,/$

%. D&'(n(#$ !& ")* M*tr(+

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4

U$ $-#/= < $ -$%*&$ .* ,*#(+ +%(&(+ ( *H)#*++*+ $-*-/$+ ./+)(+-(+ * m

&/,'$+ ;/&$+ '(#/=(,-$/+ * n (&,$+ ;/&$+ *#-/$/+ D/=*(+ $++/ * $ $-#/= )(++/

(#.* m, n&+* (#.* m)(# n

E,&)-:

ai , j Matriz mpor n

n Colunas

m Linhas

[a

1 ,1 a1 , 2 a1, 3 a1 ,na

2 ,1 a2 , 2 a2, 3 a2 ,n

am , 1 am , 2 am ,3 am, n]

* / M$-#/= ( / E&**,-( &/,'$ 0 / E&**,-( (&,$

. A-r&2&nt*#$ !& ")* M*tr(+

P(.*(+ #*)#*+*,-$# $ $-#/= $( (&($# +*+ .$.(+ *,-#* )$#,-*+*+ ( (&'*-*+ Q

E,&)-2:

A (1 2 0 8 7 ) A < $ $-#/= .* $ +J &/,'$ * /,( (&,$+ ( +*$ 1H5

B (1

2

0

x

8) B < $ $-#/= .* /,( &/,'$+ * $ +J (&,$ ( +*$ 5H1

C

(

1 2 64 0 7

3 8 9

) C < $ $-#/= .* -#+ &/,'$+ * (&,$+ ( +*$ 3H3

j

i

7/26/2019 Atps - Matrizes e Determinantes

5/17

5

3. T(-2 !& )*tr(+&2

3.1 M*tr(+ (n4*

T* ( ,(* .* $-#/= &/,'$ $*&$ * )(++/ $)*,$+ $ &/,'$ /,.*)*,.*,-* .* $& +*$ (

,*#( .* (&,$+

E,&)-2:

A (a11 a12 a13 an ) < $ $-#/= .* (#.* )(# , 1 H ,

B / (4 5 6 7 7 ) < $ $-#/= .* (#.* )(# -#+ 1 H 5

C 5 1< $ $-#/= .* (#.* )(# .(/+ 1 H 2

3.% M*tr(+ 5"n*

T* ( ,(* .* $-#/= (&,$ $*&$ * )(++/ $)*,$+ $ (&,$ /,.*)*,.*,-* .* $&

+*$ ( ,*#( .* &/,'$+

E,&)-2:

A / (a

11

a12

:

am) < $ $-#/= .* (#.* H 1

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6/17

6

B /

(

1

5

6

0

9

) < $ $-#/= .* (#.* 5 H 1

3. M*tr(+ 6"*!r*!*

-(.$ $-#/= * )(++/ ( *+( ,*#( .* &/,'$+ < .* (&,$+

E,&)-2:

A (2 35 8) $-#/= .* (#.* 2 H 2

B (2 3 2

2 11 4

5 1 3) $-#/= .* (#.* 3 H 3

3..1 D(*7n* Pr(n5(-*

P(.*(+ .*;/,/# (( ./$(,$& )#/,/)$& .* $ $-#/= $.#$.$ (( $ ,/?( .(+

*&**,-(+ .( $,-( +)*#/(# *+*#.( $( $,-( /,;*#/(# ./#*/-(E,&)-2:

A / (8 07 4)

B / (8 0 1

7 4 5

3 9 2)

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7/17

7

3..% D(*7n* S&5"n!8r(*

$ ,/?( .(+ *&**,-(+ .( $,-( +)*#/(# ./#*/-( $( $,-( /,;*#/(# *+*#.(

E,&)-2:

A / (8 07 4)

B / (8 0 1

7 4 5

3 9 2)3.3 M*tr(+ N"* " Z&r

M$-#/= * * -(.(+ (+ *&**,-(+ +?( ,&(+ < #*)#*+*,-$.( )(# 0 H , /,.*)*,.*,-* .(

,*#( .* &/,'$+ * (&,$+

E,&)-2:

A (0 00 00 0

) $-#/= .* (#.* 3 H 3

B (0 0 0

0 0 0

) $-#/= .* (#.* 2 H 3

3.9 M*tr(+ I!&nt(!*!& " Un(t8r(*

M$-#/= /.*,-/.$.* *&$ -* * +*# $.#$.$ * (+ *&**,-(+ * )*#-*,*#* ./$(,$&

)#/,/)$& .** +*# /$/+ $ 1 ( +*$ (,.* / * ( #*+-$,-* .(+ *&**,-(+ /$/+ $ =*#(

I,./$+* $ $-#/= /.*,-/.$.* .* (#.* , )(# I, ( +/)&*+*,-* )(# I

E,&)-2:

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8

I 2 [1 00 1 ] < $ $-#/= /.*,-/.$.* .* (#.* 2

I 3 (1 0 00 1 00 0 1

) < $ $-#/= /.*,-/.$.* .* (#.* 3

I 4 (1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1) < $ $-#/= /.*,-/.$.* .* (#.* 4

3. M*tr(+ Tr*n2-2t* $,.( +* -#($ .* ;(#$ (#.*,$.$ (+ *&**,-(+ .* +$+ &/,'$+ )(# (&,$+ ( .$+ (&,$+

)(# &/,'$+

E,&)-2:

A (a bc d ) A t (a cb d )

B =(0 4 6 9 )

Bt

(0

4

6

9)

3.; M*tr(+ S()

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9/17

9

S*,.( A $ $-#/= $.#$.$ * +$ $-#/= -#$,+)(+-$ At

&(( A +*# $ $-#/= +/?( A At

E,&)-:

A (2 0 21 1 20 3 0

) A t (2 1 00 1 32 2 0

)

3.= M*tr(+ Ant(>S()

7/26/2019 Atps - Matrizes e Determinantes

10/17

10

A (2 3 40 4 60 0 8

)

3.1@ M*tr(+ Tr(*n7"*r In'&r(r

P(.*(+ ./=*# * $ $-#/= -#/$,&$# /,;*#/(# < $,.( (+ *&**,-(+ $/$ .$ ./$(,$&

)#/,/)$& +?( ,&(+ ( +*$ 0

E,&)-:

A ( 2 0 06 4 05 1 8)

3.11 M*tr(+ O-2t*P$#$ *,(,-#$# $ $-#/= ()(+-$ .* $&*# $-#/= %$+-$ -#($# (+ +/,$/+ .(+ #*+)*-/(+

*&**,-(+

E,&)-:

A

( 2 3 48 1 6

9 5 7

) A

(2 3 4

8 1 69 5 7

)

9. O-&r*#&2 5) M*tr(+&2

9.1 A!(#$ & S"tr*#$ !& M*tr(+&2

7/26/2019 Atps - Matrizes e Determinantes

11/17

11

E++$ ()*#$>?( +J < )(++*& $,.( $+ $-#/=*+ )(++* $ *+$ (#.* * $+ ()*#$>*+ +?(

;*/-$+ ( (+ *&**,-(+ .* *+$ )(+/>?( (%*.**,.( $+ #*#$+ .* +/,$&

E,&)-: A!(#$

A( 2 34 7) +B=(1 52 8) ( 2 +(1) 3 +54 +2 7 +(8)) ( 1 82 1)

E,&)-: S"tr*#$

A( 2 34 7) B=(1 52 8) ( 2 (1) 3 54 2 7 (8)) ( 3 26 15 )

9.% M"t(-(5*#$ !& M*tr(+&2

P$#$ &-/)&/$#(+ .$+ $-#/=*+ $ (,./>?( ,**++#/$ < * ( ,*#( .* (&,$+ .$)#/*/#$ $-#/= +*$ /$& $( ,*#( .* &/,'$+ .$ +*,.$ $-#/= @$,.( /++( ((##* $

$-#/= )#(.-( -*# ( ,*#( .* &/,'$+ .$ )#/*/#$ $-#/= * ( ,*#( .* (&,$+ .$ +*,.$

$-#/= E++* )#(.-( .** +*# ;*/-( &-/)&/$,.( $.$ &/,'$ .$ )#/*/#$ $-#/= )(# $.$

(&,$ .$ +*,.$ $-#/= *&**,-( )(# *&**,-( -($,.( /.$.( ( +*+ +/,$/+

Cn!(#$ n&5&228r(* -*r* )"t(-(5*#$ !& M*tr(+&2.

E,&)-2:

A2x 3 B3x 3 )(++*& )(/+ ( , .* (&,$+ .$ 1 < /$& $( , .* &/,'$+ .$ 2

B3x 3 A2x 3 N?( < )(++*& )(/+ ( , .* (&,$+ .$ 1 < ./;*#*,-* $( , .* &/,'$+ .$ 2

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12

S*,.( $ $-#/= A(1 2 34 5 6) * B (198) .*-*#/,*:

A2x 3 B 3x 1

(1 2 34 5 6) H (198) (1.1+2.9+3.84.1+5.9+6.8) (4397)

B 3x 1 A 2x 3 N?( < )(++*& )(/+ ( , .* (&,$+ .$ 1 < ./;*#*,-* $( , .* &/,'$+ .$

2

S*,.( $ $-#/= A[1 34 2] * B [1 23 4] .*-*#/,*:

AB [1 34 2] [1 23 4] [ (1 ) .1+3.3 (1 ) .2+3.44.1+2.3 4.2+2.4] [ 8 1010 16]

. D&t&r)(n*nt&2

C(( /(+ $-#/= $.#$.$ < $ * -* ( *+( ,*#( .* &/,'$+ * .* (&,$+

A -(.$ $-#/= $.#$.$ *+- $++(/$.( ,*#( $( $& .$(+ ( ,(* .* .*-*#/,$,-*

P(.*(+ ./=*# * ( .*-*#/,$,-* .* $ $-#/= $.#$.$ < ( +* $&(# ,?(: R*)#*+*,-$(+ ( .*-*#/,$,-* .* $ $-#/= *,-#* .$+ %$##$+ *#-/$/+ *

,?( - ( +/,/;/$.( .* J.&(

E,&)-2:

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13/17

13

A [ 5 ] .*- A 5 ( |5| 5

M [3 ] .*- M 3 ( |3| 3

.1 D&t&r)(n*nt&2 !& 1 r!&)

@$,.( $ $-#/= )(++/ $)*,$+ *&**,-( ( )(++/ $)*,$+ $ &/,'$ * $ (&,$

./=*(+ * *++$ $-#/= < .* (#.* 1

P(.*(+ (,&/# * ( .*-*#/,$,-* .* (#.* 1 -*# ( +* $&(# ,$ *,-#* ( )#(.-( .(+

*&**,-(+ .$ ./$(,$& )#/,/)$& * ( )#(.-( .(+ *&**,-(+ .$ ./$(,$& +*,.#/$

E,&)-2:

[ 1 2

5 3 ].*- A 10 3 3 10 3 Z 10 7

S*,.( M |2 34 5| -*(+:

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14/17

14

.*- M |2 34 5| 25 43 10 12 .*- M 2

.% M*tr(+ !& !& r!&)

O .*-*#/,$,-* .* $ $-#/= .* 3 (#.* < $&&$.( .$ +*/,-* ;(#$ )#/*/#(#*)#*+*,-$(+ *++$ $-#/= * ;(#$ .* .*-*#/,$,-* * #*)*-/(+ $+ .$+ )#/*/#$+ (&,$+

(,;(#* *H*)&( $%$/H( * < (,'*/.$ (( ;(#&$ .* S$##+

E,&)-2:

[

a11

a12

a13

a21

a22

a23

a31

a32

a33

]

a11

a12

a21

a22

a31

a32

=

a

) +

a

)

S*,.( A ( 1 4 2

3 0 1

1 2 3) -*(+:

.*-A ( 1 4 23 0 11 2 3) 1 4

3 0

1 2

.*- A 103Z411Z232 102Z211Z334

.*- A 0Z4Z12 0Z2 Z36

7/26/2019 Atps - Matrizes e Determinantes

15/17

15

.*- A 8 34

.*- A 42

S*,.( B (1 2 3

4 6 6

7 8 9) -*(+:

.*-B (1 2 34 6 67 8 9

)1 24 67 8

.*- B 169Z267Z348 367Z168Z249

.*- B54Z 84Z96 126 Z 48 Z 72

.*- B234 246

.*- B 12

;. Cn5"2$

MATRIZ: P(.*(+ .*;/,/# $ $-#/= (( +*,.( $ -$%*&$ (#$,/=$.$ * (&,$+ *

&/,'$+ +*,.( #*)#*+*,-$.$ .* ;(#$ m&/,'$+ * n(&,$+ ( +%(&(+ ( *H)#*++*+ +*,.(

* $.$ .(+ /-*,+ .* $ $-#/= < '$$.( .* *&**,-(

DETERMINANTES: P(.*(+ .*;/,/# .*-*#/,$,-* (( +*,.( ,*#( ( $

;,>?( $++(/$.( $ $ $-#/= $.#$.$ $&&$.( .* $(#.( ( #*#$+ *+)*;/$+

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16

=. B((7r*'(*

1 '--):[[\\\%#$+/&*+(&$([$-*$-/$[.*-*#/,$,-*+1'-

2 '--):[[\\\/,;(*+(&$([$-*$-/$[$-#/=-#$,+)(+-$[

3 '--):[[\\\/,;(*+(&$([$-*$-/$[$-#/=*+[

4 '--):[[\\\,.(*.$$(([$-*$-/$[.*-*#/,$,-*$-#/=(#.*12(3'-

5 '--):[[\\\(($&&$#(%#[(( $&&$#[*+(&$[(* *$-#/=+/*-#/$

6 '--):[[\\\,.(*.$$(([$-*$-/$[.*-*#/,$,-*$-#/=+/*-#/$'-

7 '--):[[\\\;*,*+)%#[*H-*,+$([-*/$[$&$+[A,$12$(+-(M$-#/=*+).;

8 '--):[[\\\;*,*+)%#[*H-*,+$([-*/$[$&$+[A,$12$(+-(M$-#/=*+).;

9 '--):[[+&/.*)&$*#(%#[+&/.*[1626192[

10 '--):[[\\\,.(*.$$(([$-*$-/$[$-#/=$.#$.$'-

11 R*;*#,/$ %/%&/(#;/$: S(=$ K($/# R(%*#-( .* N(( (&'$# $-*-/( 1 *. S?(

P$&(: FTD 2010 C(&*>?( O&'$#] V2

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17

12 '--):[[\\\)#(;$#.([$#./$+[-/#$-*/$)')^/.12

13 \\\+($-*$-/$(%#[**./([$-#/=*+[$-#/=*+2)')

14 L&/#(-*H-(: S-*/,%#' F W/,-*#&* P &*%#$ L/,*$# * G*(*-#/$ A,$&-/$ 2 *./>?(S?( P$&(: P*$#+(, E.$-/(, 2007 PLTA,'$,*#$ E.$/(,$& S*+-*+ %/%&/(#;/$+:

15 A&*%#$ L/,*$# C( A)&/$(*+ C'#/+ R(##*+ !(\$#. A A,-(, 8 *./>?( P(#-(

A&*#*: BOO_MAN 2001