Calculo vetorial

Cálculo Vetorial e Álgebra Linear
Lista 2
1° Semestre de 2013 – Prof. Claudio H. Asano

1

Matrizes
1 −1 5 2 3 2

1.1 São dadas as matrizes A =

,B=

1 4

1

2 4 −2

eC=

2 1 1 −3 1 5

. Calcule:

(a) A − 2B + 3C.
Resp: 5 −11 −6 −2 6 21

(b) 3B − 4(A − 2C).
Resp: 15 −26 24 −9 8 26

1.2 São dadas as matrizes A =

2 −3 1 −2

,B=

−5 −3 −1 −3

eC=

−1 −2 −2 −4

. Calcule:

(a) AB − C.
Resp: −6 −1 5 7

(b) A − BC.
Resp: −9 −6 −25 −16

1.3 São dadas as matrizes A =

4 −3 2 3

,B=

2 −4 −2 −4

eC=

3 −1 −5 −3

. Calcule:

(a) A2 + B 2 .
Resp: 22 18 −13 27

(e) A2 − 2AB + B 2 .
Resp: −6 22 −5 67

(b) (A + B)2 .
Resp: 36 0 −35 1

(f) (A + B)(A − B).
Resp: −16 −4 −43 −7

(c) A2 + 2AB + B 2 .
Resp: 50 −21 14 −13

(g) (A − B)(A + B).
Resp: 12 −15 24 −35

(d) (A − B)2 .
Resp: 8 36 9 53

(h) A2 − B 2 .
Resp: −2 −29 10 −21

1

1.4 Dadas as matrizes



2 2 2

 e

  B= 

−1 0 2



  A= 3 2 1    1 2 1 calcule a diferença AB − BA.
Resp: 

 4 3 1   1 4 3

8 −9

12 −6

12



 AB − BA =  −12 −6 

 −1   −2

 1.5 Dadas as matrizes A = 3 4 −4 −3 ,B= −4 1 −3 −3 1 −3

−1 4 −2



 e C =  −3 3  2 2 (c) det C

 2 , calcule:  0

(a) AB
Resp: −24 7 −21 21

(b) A−1
3 −7 4 7 4 −7 3 7

Resp: 44 Resp:

25 −7

1.6 Calcule o determinante das matrizes quadradas abaixo:    3 2 1 −3 1 0     2 2 2   4 2 1 (a)  (c)   5 2 1 1 5 3
Resp: 4 Resp: −14

   



4 3 1





1 2 3



  (b)  −1 0 1    1 4 3
Resp: −8

  (d)  4 5 6    7 8 9
Resp: 0

−3 −4 3 2    −3 −3 −2 0    1.7 Dada a matriz A =  , calcule o seu determinante.  −2 1 −4 −3    0 2 −2 −3
Resp: det A = 35





    1.8 Calcule o determinante da matriz   
Resp: −217

−5 −3 3 4 4 2

2 4



 5 4   . 3 −1 3   4 1 2

2



5

0

0

0 0

     , calcule o seu determinante.