Matrices Semana 5(1)
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Transcript of Matrices Semana 5(1)
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8/17/2019 Matrices Semana 5(1)
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ALGEBRA LINEAL
Ejercicios Propuestos Semana 5
Matrices
TUTOR(A)
Mirea G!me"
Presenta#o Por
Ar$e Buritic% Prieto
&!#i'o *+,*-
Grupo ,../.01*0*
UNA2
Uni$ersi#a# Naciona3 A4ierta a 2istancia
L4ano 6 To3ima
*.,+
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8/17/2019 Matrices Semana 5(1)
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E7ER&I&IOS PROPUESTOS
,8 Encuentre 3a matri" in$ersa #e A=
(
1 1 1 1
1 2 −1 21 −1 2 1
1 3 3 2
) 9acien#o uso #e3 m:to#o #e
Gauss67or#%n 3ue'o por e3 m:to#o #e 3os #eterminantes8
Recuer#e ;ue< A−1=
1
Det A∗ AdjA
M:to#o Gauss67or#%n
| 1 1 1 1
1 2 −1 21 −1 2 1
1 3 3 2 |1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1|⟹ f 2−f 1
| 1 1 1 1
0 −1 −2 11 −1 2 1
1 3 3 2|
1 0 0 0
−1 1 0 00 0 1 0
0 0 0 1|⟹ f 3−f 1
| 1 1 1 1
0 −1 −2 10 −2 1 0
1 3 3 2|
1 0 0 0
−1 1 0 0−1 0 1 0
0 0 0 1|⟹ f 4−f 1
|
1 1 1 1
0 −1 −2 10 −2 1 0
0 2 2 1
|
1 0 0 0
−1 1 0 0
−1 0 1 0−1 0 0 1
|⟹−f
2
| 1 1 1 1
0 1 2 −10 −2 1 0
0 2 2 1|
1 0 0 0
1 −1 0 0
−1 0 1 0−1 0 0 1
|⟹ f 1−f 2
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|1 0 −1 20 1 2 −10 −2 1 0
0 2 2 1|
0 1 0 0
1 −1 0 0−1 0 1 0−1 0 0 1
|⟹ f 3+2 f 2
|1 0 −1 20 1 2 −10 0 5 −2
0 2 2 1|
0 1 0 0
1 −1 0 01 −2 1 0−1 0 0 1
|⟹ f 4−2 f 2
|1 0 −1 2
0 1 2 −10 0 5 −20 0 −2 3|
0 1 0 0
1 −1 0 01 −2 1 0−3 2 0 1|
⟹1
5f 3
|1 0 −1 20 1 2 −1
0 0 1 −2
5
0 0 −2 3|
0 1 0 0
1 −1 0 01
5
−25
1
50
−3 2 0 1|⟹ f 1+ f 3
| 1 0 0
8
5
0 1 2 −1
0 0 1 −2
5
0 0 −2 3|
1
5
3
5
1
5 0
1 −1 0 01
5
−25
1
50
−3 2 0 1|⟹ f 2−2 f 3
| 1 0 0
8
5
0 1 0 −1
5
0 0 1 −2
5
0 0 −2 3| 1
5
3
5
1
50
3
5
−15
−25
0
1
5
−25
1
50
−3 2 0 1 |⟹ f 4+2 f 3
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| 1 0 0
8
5
0 1 0 −1
5
0 0 1 −2
5
0 0 0 11
5 |
1
5
3
5
1
5 0
3
5
−15
−25
0
1
5 −2
5
1
50
−135
6
5
2
51
|⟹
5
11f
4
| 1 0 0
8
5
0 1 0 −1
5
0 0 1 −2
5
0 0 0 1 |
1
5
3
5
1
5 0
3
5
−15
−25
0
15
−25
15
0
−1311
6
11
2
11
5
11|⟹ f
3+
2
5f
4
| 1 0 0
8
5
0 1 0 −1
5
0 0 1 0
0 0 0 1 |
1
5
3
5
1
5 0
3
5
−15
−25
0
−311
−211
3
11
2
11
−1311
6
11
2
11
5
11|⟹ f
2+
1
5
f 4
|1 0 0
8
5
0 1 0 0
0 0 1 0
0 0 0 1|
1
5
3
5
1
5 0
4
11
−111
−411
1
11
−311
−211
3
11
2
11
−1311
6
11
2
11
5
11|⟹ f
1−
8
5
f 4
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8/17/2019 Matrices Semana 5(1)
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|
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
|
23
11
−311
−111
−811
4
11
−111
−411
1
11
−3
11 −2
11
3
11
2
11
−1311
6
11
2
11
5
11|En =orma e;ui$a3ente A
−1= 1
11
|
23 −3 −1 −84 −1 −4 1−3 −2 3 2−13 6 2 5
|M:to#o por 3os #eterminantes
A−1=
1
Det A∗ AdjA
A=( 1 1 1 1
1 2 −1 21 −1 2 1
1 3 3 2 )2eterminante
| A|=C 11+C +C
31+C
41
1 (−1 )1+1| 2 −1 2−1 2 1
3 3 2|+1 (−1 )2+1|
1 1 1
−1 2 13 3 2
|+1 (−1 )3+1|1 1 1
2 −1 23 3 2
|+1 (−1 )4+1| 1 1 1
2 −1 2−1 2 1|
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8+(−3 )+(−6 )−( 12+6+2 )−3
4+3+(¿)¿−2
6+3+(¿)¿
¿−¿¿¿
−1
[ ( (−2 )+6+6 )−( (−3 )+6+4 ) ]−[ (¿+(−2)+4 )− (1+4+2 )]
¿ [ 8−3−6−12−6−2 ]−[ 4+3−3−6−3+2 ]+¿
[−2+6+6+3−6−4 ]−[−1−2+4−1−4−2]
¿ (−21 )−(−3 )+ ( 3 )−(−6)
¿−21+3+3+6=−9
A#junta
Adj=[+
| 2 −1 2−1 2 1
3 3 2
|−
|1 −1 21 2 1
1 3 2
|+
|1 2 2
1 −1 11 3 2
|−
|1 2 −11 −1 21 3 3
|−| 1 1 1−1 2 1
3 3 2|+|1 1 11 2 1
1 3 2|−|1 1 11 −1 1
1 3 2|+|1 1 11 −1 2
1 3 3|
+|1 1 1
2 −1 23 3 2
|−|1 1 1
1 −1 21 3 2
|+|1 1 1
1 2 2
1 3 2|−|
1 1 1
1 2 −11 3 3
|−
|
1 1 1
2 −1 2
−1 2 1
|+
|
1 1 1
1 −1 2
1 2 1
|−
|
1 1 1
1 2 2
1 −1 1
|+
|
1 1 1
1 2 −1
1 −1 2
|
][ 8−3−6−12−6−2 ]−[ 4−1+6−4−3+2 ]+ [−2+2+6+2−3−4 ]−[−3+4−3−1−6−6]
−[ 4+3−3−6−3+2 ]+ [ 4+1+3−2−3−2 ]−[−2+1+3+1−3−2 ]+[−3+2+3+1−6−3]
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[−2+6+6+3−6−4 ]−[−2+2+3+1−6−2 ]+ [ 4+2+3−2−6−2 ]−[6−1+3−2+3−3 ]
−[−1−2+4−1−4−2 ]+[−1+2+2+1−4−1 ]−[2+2−1−2+2−1 ]+[4−1−1−2−1−2]
[−21
]− [4
]+[1
]−[−15
]
−[−3 ]+[ 1 ]−[−2 ]+[−6]
[ 3 ]−[−4 ]+ [−1 ]−[6]
−[−6 ]+[−1 ]−[2 ]+[−3]
Adj=
|−21 −4
3 1
1 15
2 −63 4
6 −1−1 −6−2 −3|
A−1=
1
Det A∗ AdjA
A−1=
1
−9∗
|−21 −4
3 1
1 15
2 −63 4
6 −1 −1
−6
−2 −3
| Adj=
|
7
3
4
9
−13
1
9
−19
−53
−29
2
3
−13
−49
−23
1
9
1
9
2
3
2
9
1
3
|*8 >a33e 3a matri" esca3ona#a #e 3a matri" A 3ue'o #etermine si es una matri"in$erti43e8
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8/17/2019 Matrices Semana 5(1)
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A=[ 2 −1 4−1 0 519 7 3
] Esca3ona#a
A=[ 2 −1 4−1 0 519 7 3
]⟹12∗f 1
A=[ 1 −1
2 2
−1 0 519 7 3
]⟹ f 2+ f 1
A=[ 1
−12
2
0 −1
27
19 7 3
]⟹ f 3−19 f 1
A=
[1 −1
22
0 −12
7
0 33
2−35]⟹−2∗f 2
A=
[
1 −1
22
0 1 −14
0 33
2−35
]⟹ f
3−
33
2f
2
A=[1 −1
2 2
0 1 −140 0 196
]⟹ 1196∗f 3
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A=[1 −1
2 2
0 1 −140 0 1
]In$erti43e
A=[ 2 −1 4−1 0 519 7 3
] A=
| 2 −1 4−1 0 519 7 3
|1 0 0
0 1 0
0 0 1
|⟹
1
2
∗f 1
A=| 1 −1
22
−1 0 519 7 3
|1
20 0
0 1 0
0 0 1|⟹ f 2+ f 1
A=
| 1
−12
2
0 −1
27
19 7 3|1
20 0
1
21 0
0 0 1|⟹ f
3−19 f 1
A=
|
1 −1
22
0 −1
27
0 33
2
−35
|
1
20 0
1
21 0
−19
2
0 1
|⟹−2∗f
2
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A=|1 −1
22
0 1 −14
0 33
2−35|
1
20 0
−1 −2 0−19
20 1|⟹ f 3−332 f 2
A=|1 −1
22
0 1 −140 0 196
| 1
20 0
−1 −2 07 33 1
|⟹ 1196∗f 3
A=
|1 −1
22
0 1 −140 0 1
|
1
20 0
−1 −2 01
28
33
196
1
196|
