atps - matrizes e determinantes
TRANSCRIPT
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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(
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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*-*#/,$,-*+ 3 O#.* 13
7 C(,&+?(
8 B/%&/(#$;/$
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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
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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
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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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
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()
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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
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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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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
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
.*- M |2 34 5| 25 43 10 12 .*- M 2
.% M*tr(+ !& !& r!&)
O .*-*#/,$,-* .* $ $-#/= .* 3 (#.* < $&&$.( .$ +*/,-* ;(#$ )#/*/#(#*)#*+*,-$(+ *++$ $-#/= * ;(#$ .* .*-*#/,$,-* * #*)*-/(+ $+ .$+ )#/*/#$+ (&,$+
(,;(#* *H*)&( $%$/H( * < (,'*/.$ (( ;(#&$ .* S$##+
E,&)-2:
[
a11
a12
a13
a21
a22
a23
a31
a32
a33
]
a11
a12
a21
a22
a31
a32
=
a
) +
a
)
S*,.( A ( 1 4 2
3 0 1
1 2 3) -*(+:
.*-A ( 1 4 23 0 11 2 3) 1 4
3 0
1 2
.*- A 103Z411Z232 102Z211Z334
.*- A 0Z4Z12 0Z2 Z36
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.*- A 8 34
.*- A 42
S*,.( B (1 2 3
4 6 6
7 8 9) -*(+:
.*-B (1 2 34 6 67 8 9
)1 24 67 8
.*- B 169Z267Z348 367Z168Z249
.*- B54Z 84Z96 126 Z 48 Z 72
.*- B234 246
.*- B 12
;. Cn5"2$
MATRIZ: P(.*(+ .*;/,/# $ $-#/= (( +*,.( $ -$%*&$ (#$,/=$.$ * (&,$+ *
&/,'$+ +*,.( #*)#*+*,-$.$ .* ;(#$ m&/,'$+ * n(&,$+ ( +%(&(+ ( *H)#*++*+ +*,.(
* $.$ .(+ /-*,+ .* $ $-#/= < '$$.( .* *&**,-(
DETERMINANTES: P(.*(+ .*;/,/# .*-*#/,$,-* (( +*,.( ,*#( ( $
;,>?( $++(/$.( $ $ $-#/= $.#$.$ $&&$.( .* $(#.( ( #*#$+ *+)*;/$+
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=. B((7r*'(*
1 '--):[[\\\%#$+/&*+(&$([$-*$-/$[.*-*#/,$,-*+1'-
2 '--):[[\\\/,;(*+(&$([$-*$-/$[$-#/=-#$,+)(+-$[
3 '--):[[\\\/,;(*+(&$([$-*$-/$[$-#/=*+[
4 '--):[[\\\,.(*.$$(([$-*$-/$[.*-*#/,$,-*$-#/=(#.*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
http://www.brasilescola.com/matematica/determinantes-1.htmhttp://www.infoescola.com/matematica/matriz-transposta/http://www.comocalcular.com.br/comohttp://www.feg.unesp.br/extensao/teia/aulas/Ana12agosto-Matrizes.pdfhttp://www.feg.unesp.br/extensao/teia/aulas/Ana12agosto-Matrizes.pdfhttp://slideplayer.com.br/slide/1626192/http://www.brasilescola.com/matematica/determinantes-1.htmhttp://www.infoescola.com/matematica/matriz-transposta/http://www.comocalcular.com.br/comohttp://www.feg.unesp.br/extensao/teia/aulas/Ana12agosto-Matrizes.pdfhttp://www.feg.unesp.br/extensao/teia/aulas/Ana12agosto-Matrizes.pdfhttp://slideplayer.com.br/slide/1626192/ -
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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
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