Exerc Cios Resolvidos De Lgebra Linear 1

Instituto Superior T´ecnico
Departamento de Matem´atica
´
Sec¸c˜ao de Algebra e An´alise

´
Alguns Problemas e Exames Resolvidos de Algebra
Linear
LEAmb, LEAN, LEMat, LQ, MEBiol, MEQ
1o Semestre 2008/2009
Prof. Paulo Pinto http://www.math.ist.utl.pt/∼ppinto/ Conte´ udo 1 Alguns problemas resolvidos
1.1 Resolu¸c˜ao de alguns exames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Exames sem resolu¸c˜ao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Consultar exames em: http://www.math.ist.utl.pt/∼ppinto/AL/exames.html 22

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Alguns problemas resolvidos

1.1 O sistema linear

 x+z =3





 x + 2y + 2z = 6






3y + 3z = 6

na forma matricial ´e

Consideremos ent˜ao a

1 0

 1 2
0 3



   
1 0 1 x 3

   
 1 2 2  y  =  6 .
0 3 3 z 6 matriz aumentada e o consequente m´etodo de elimina¸c˜ ao 



1 0 1 | 3
1 0 1
1 | 3




2 | 6  −→  0 2 1 | 3  3−→  0 2 1
−L1 +L2
− 2 L2 +L3
0 3 3 | 6
0 0 32
3 | 6

Logo,



x+z =3 x=2 











2y + z = 3 ⇔ y=1 









 3

3 z = 1. z =
2
2

1.2 O sistema linear

´e equivalente a

de Gauss:

| 3

| 3 .
| 32


3z − 9w = 6






5x + 15y − 10z + 40w = −45





 x + 3y − z + 5w = −7






0 0
3
−9 


 5 15 −10 40  

1 3 −1
5

x y z w 



6

 

 =  −45  .

−7

Consideremos ent˜ao a matriz aumentada e o consequente m´etodo de elimina¸c˜ ao de Gauss:




0 0
3
−9 |
6
1 3 −1 5 | −7




 5 15 −10 40 | −45  −→  1 3 −2 8 | −9  −→
L1 ↔L3
−L1 +L2
1
1 3 −1
5 | −7
0 0 3 −9 | 6
L
2
5




1 3 −1 5 | −7
1 3 −1 5 | −7




−→  0 0 −1 3 | −2  −→  0 0 −1 3 | −2  .
3L2 +L3
0 0 3 −9 | 6
0 0 0 0 | 0
2

Logo,



 x = −3y − 2w − 5



 x + 3y − z + 5w = −7



⇔
−z + 3w = −2




z = 3w + 2.

As inc´ognitas y e w s˜ao livres e as inc´ognitas x e z s˜ ao n˜ao livres.

 
−3y − 2w − 5 x  y   y 
 
X=
=
 z  
3w + 2 w w

A solu¸c˜ ao geral do