1.-Álgebra matricial

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CAPITULO I : ALGEBRA MATRICIAL22 SEPTIEMBRE 2009

UNIVERSIDAD CENTRAL DEL ECUADOR

FACULTAD DE INGENIERÍA

ESCUELA CIVIL

ESTRUCTURAS III

Nombre: Conrado Díaz María Beln Fe!"a: ##de SC)r*o: Se$%&mo Se+)ndo

,- CA.ITULO , DEFINICI/N DE MATRI

m x n.

[A]= mxn… … … …

2. TIPOS DE MATRICESDentr !e "#$ e$tr%&t%r#$ %t'"'(#m$ &'ert t') !e m#tr'&e$.

2.1 MATRIZ CUADRADA

E$ #*%e""# *%e t'ene e" m'$m n+mer !e ,"#$ *%e !e &"%mn#$- e$ !e&'r m = n.

2 /

1 3 4 m=

[B]= 9 0 2 n=

/ 1 m=n

2.2 MATRIZ COLUMNAE$ %n# m#tr'( *%e $" t'ene %n# &"%mn#- e$ !e&'r- n =

1 m=1

[C]= 9 n=

/

3

2.3 MATRIZ FILAE$ %n# m#tr'( *%e $" t'ene %n# ,"#- e$ !e&'r m =

m=

[5]= 9 0 2 n=

2.4 MATRIZ DIAGONALE$ %n# m#tr'( &%#!r#!#- en "# *%e t!$ "$ e"ement$ n )ertene&'ente$ # "# !'#6n#" )r'n

Se ""#m# m#tr'( # t! &n7%nt re&t#n6%"#r !e e"ement$ a'7 !'$)%e$t$ en m ,"#$ 8 n &"%mn#

#

#2

#/

#n

#2

#22

#2/

#2n

#/ #/2 #// #/n

#m

#m2

#m/

#mn


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$n n%"$.

0 0 0 0 0

0 20 0 0 0 m=1

[D]= 0 0 /0 0 0 n=1

0 0 0 0 0 m=n

0 0 0 0 10

2.5 MATRIZ NULAE$ #*%e""# *%e t!$ $%$ e"ement$ $n 0

0 0 0

[]= 0 0 0

0 0 0

2.6 MATRIZ IDENTIDADE$ %n# m#tr'( e$"#r &n "$ e"ement$ !e "# !'#6n#" )r'n&')#" '6%#"e$ #

0 0 0 0

0 0 0 0 m=1

[I]= 0 0 0 0 n=1

0 0 0 0 m=n

0 0 0 0

2.7 MATRIZ TRIANGULAR SUPERIORE$ &%#n! "$ e"ement$ *%e e$tn )r !e;#7 !e "# !'#6n#" )r'n&')#" $n &er$.

2 /

0 3 4 m=

[S]= 0 0 2 n=

0 0 0 m=n

2.8 MATRIZ TRIANGULA INFERIORE$ &%#n! "$ e"ement$ *%e e$tn )r en&'m# !e "# !'#6n#" )r'n&')#" $n &er$.

0 0 0

2 0 0 m=

[T]= / 0 n=

1 3 4 m=n

2.9 MATRIZ TRANSPUESTA

20 / 20 22 1 4

[A]= 3 / / 1

Se ""#m# tr#$)%e$t# !e A- # "# re)re$ent#&'


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1 0 9 4

m=n

2.10 MATRIZ SIMÉTRICA

2 / 1

2 3 4 m=n

[A]= / 3 / 0

1 4 0 9

3. OPERACIONES CON MATRICES

3.1 TRANSPOSICIÓN DE MATRICES

1 4 0 1 1

[E]= 1 2 0 2

9 2 4

/ 3 1 22 0 0

1 4 0

1 2 0

9 2

/ 3 1 22

3.2 SUMA RESTA DE MATRICES

EEMPLO

/ 1 4 1

[>]= 0 2 ? [@]= 4 9 [>] ?[@]=

3 / 9 3 / 2

EEMPLO

14 90 1 2

[M]= 0 1 0 ? []= / 1 ? [O]=

1 0 0 2

3 22

[M] ?[]?[O]= 1 2

3 1 2

EEMPLO

Un# )r)'e!#! !e "# m#tr'( tr#n$)%e$t# e$ "# $'6%'ente : E tt = E

[Et]=

[Et]t=


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9 1 1 4 1

[>]= 2 / [@]= 4 9 [>] [@]=

3 3 3 / 2

EEMPLO

2 0 1 1 2

[M]= 20 1 / []= 1 ? [O]=

1 0 0 2

1 3 /

[M] []?[O]= / 21 /

3 2

3.3 PRODUCTO DE UNA MATRIZ POR UN ESCALAR

EEMPLO

1 9 / 0 10

2 0 20 0

[A]= / 3 1 0[S]= /0 30

4 2 0 40

EEMPLO

[]=2[C]?/[L]

/ 1 4 1

[C]= 0 2 [L]= 4 9

3 / 9 3 / 2

2 0 2 2 1

2[C]= 20 2 2 /[L]= 1 / 23 []=

4 2 9 /

P!"#$%&'&%(

. [A] ?[B] = [A] ? [B] )r)'e!#! !'$tr';%t'F#2. [A] =[A] %n'!#!

3.4 PRODUCTO DE MATRICES

D#!#$ !$ m#tr'&e$ A 8 B- $% )r!%&t e$ tr# m#tr'( &%8$ e"ement$ $e ;t'enen m%"t')"

,"#$ !e A )r "#$ &"%mn#$ !e B.

EEMPLO

1 9 0 1 0

[A]= 3 1 [B]= 1 4 [A][B]=


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2 / 9 / 0

P!"#$%&'&%( . AHBHC = AHBHC

2. E" )r!%&t !e m#tr'&e$ en 6ener#" n e$ &nm%t#t'F.

/. E" )r!%&t !e m#tr'&e$ e$ !'$tr';%t'F re$)e&t !e "# $%m# !e m#tr'&e$: AHB ? C = AH

3.4 IN)ERSA DE UNA MATRIZ

EEMPLO

2 / 3 0.2/2 0.0/ 0.91

[A]= / 9 2 0.0/ 0.2 0.014

3 2 4 0.91 0.014 0.0/

P!"#$%&'&%(

3 / 0.121 0.001 0.02

[A]= 1 0.001 0.030 0.0/

/ 2 0.02 0.0/ 0.01/

9 2 0.22/ 0.99 0.01

[A]= 9 0 0.9/ 0.032 0.02

1 0.031 0.031 0.01

. InFer$# !e %n# m#tr'( &%#!r#!# [B][A]=[A][B]=[I]

0 / 0 2

[A]= / / /.1 3.1 [A][B]=[I]

/ / / 3

3.5 DETERMINANTESDentr !e "# 'n6en'er'# &'F'"- en e$tr%&t%r#$ n$ #8%!# # !eterm'n#r "# e$t#;'"'!#! !e "#$ e$

m'entr#$ m$ !e #&er*%e e" !eterm'n#nte # 0 m#$ 'ne$t#;"e $er "# m'$m# 8 m'entr#$ m#$

$er m#$ e$t#;"e

20 /

2 1 4

[A]= 3 / IDI = 101 0 9

[A]=

. [ A][A]=[A][A]=I

[A]= [A] [A]=

2. AHB =BA

/. A =A

[A]= [A]=

[B]=


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/ 0

2 9

[A]= 1 / 9 IDI = 1

0 1 /

21 /

2 1 1 4

[A]= 3 9 IDI = 190

1 0 0

UNIVERSIDAD CENTRAL DEL ECUADOR

FACULTAD DE INGENIERÍA

ESCUELA CIVIL

ESTRUCTURAS III

Nombre: Conrado Díaz María Beln Fe!"a: #1de SC)r*o: Se$%&mo Se+)ndo

,- CA.ITULO II SOLUCI/N DE ECUACIONES .OR MATRIC

Un $'$tem# !e e&%#&'ne$ "'ne#"e$- t#m;'n &n&'! &m $'$tem# "'ne#" !e e&%#&'ne$

$'m)"emente $'$tem# "'ne#"- e$ %n &n7%nt !e e&%#&'ne$ "'ne#"e$.

/x 28 ( =

2x 8 ( = 2

x 0-18 ( = 0

Cn$'$te en en&ntr#r "$ F#"re$ !e$&n&'!$ !e "#$ F#r'#;"e$ x- x2 8 x/ *%e $#t'$J#&en "

e&%#&'ne$

E" mt! m$ #)r)'#! )#r# "# m#ter'# !e e$tr%&t%r#$ $er )r me!' !e "# 'nFer$# !e %

x ;

8 = &

… … … … … !( e

EEMPLO

4 0 K 20

[]= 3 K2 = 0

#

#2

#/

#n

#2

#22

#2/

#2n

#m

#m2

#m/

#mn

[][θ]=[MJ]

[][][θ]=[][MJ]

[I][θ]=[][MJ]

[θ]=[][MJ]


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0 3 29 K/ 3

0.02 0.002 0.0001

0.002 0.02/4 0.00

0.0001 0.00 0.0/19

0.0/

= 0.14

0.20/

EEMPLO

2 1 0 0 K 3.29

[]= 1 0 K2 = 3.290 /4 3 K/ 3.29

0 0 3 2 K 3.29

0.023 0.001 0.0004 0

0.001 0.02/4 0.00 0.00

0.0004 0.00 0.024/ 0.004

0 0.00 0.004 0.001

0.239

= 0.0.24

0.2

[]=

[θ]=[][MJ]

[]=

[θ]=[][MJ]


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$%&embre #''(

&')#"

- !e %n r!en


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13

/ 0

"%mn#$.


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9

9 /

3

2 1

22

m=n

1 20

24 0

0 2

2 / 40

/ 1

2 0 0


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1 / 0

4

2 / /0

/ 1

2 2 0

90 /0

00 0

0 10

20 0

0 4 /1

/ 23 21

/1 / 4

#n! "#$

/0 93 9

33 34 1


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39 /

B ? AHC

.02/3 0.00 0.00

0.00 .001 0.00

0.00 0.00 .00/

9 2

9 0

1

0 0

0 0

0 0

r%&t%r#$

#"t e" F#"r


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$%&embre #''(

ES INVERSAS

"#$ tre$

# m#tr'(


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1.-Álgebra matricial

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