Algebra

1. Calcule o valor de m e n para que as matrizes A e B sejam iguais. A= m2 − 40 n2 + 4 6 3 eB= 2 3 41 13 6 8 3 . 5 −7 −9 0 4 1 0 9 8 1 4 6

2. Dadas as matrizes A = calcule: (a) A + B (b) A − C (c) X = 4A − 3B + 5C.

4 −1 −6

,B=

eC =

,



1 3

2 9

3





x1



 3. Efetue a multiplica¸˜o das matrizes A =  −2 −5 ca   

   7  e X =  x2 .    x3 −8

1 −2

   3 1    4. Dadas as matrizes A =  eB=  7 −4    5 9 (a) A.B (b) B.A

1 3 −5 −7 6 2 −8 3

, calcule:

5. Nos exerc´ ıcios abaixo, verifique se a matriz B ´ e    −2 −4 −6 −3 2    2 5 (a) A =  −4 −6 −6  e B =  2 − 2    −1 1 −4 −4 −2    4 5 0 9 3 4     2  e B =  −7 2 5 (b) A =  3 0   1 6 8 −6 −1 −2

a inversa da matriz A:  −3 2  3  2  −1 2   . 

6. Calcule os valores de m e n para que a matriz B seja a inversa da matriz A. A= m −2 −22 n eB= 5 22 2 9 1



5

0

6

   1 −3 −2 4     7 8  e B =  5 9   0 6 3 −8   , calcule 

  −8 0 3  7. Dadas as matrizes A =   −2 2 7  1 −1 −5 (AB)T . 8. Dadas as matirzes:

 A= 0 1 1 0 ,B= 6 9 −4 −6 ,C= 5 10 −2 −4

−1

2

6



  e D =  3 −2 −9    −2 0 3

(a) Calcule A.AT (b) Calcule B 2

(c) Calcule C 2

(d) Calcule D3    3 4 1 4 −1 3    9. Dadas as matrizes A =  −5 −2 −9  e B =  3 0 1    7 8 6 7 2 −4 (a) detA (b) detB (c) det(A + B)

  , calcule: 

10. Verifique, usando o exerc´ anterior, se det(A + B) = detA + detB. ıcio   −2 3 1 −1    0 1 2 3    11. Calcule o determinante da matriz A =  .  1 −1 1 −2    4 −3 5 1 5 12. Resolva a equa¸˜o ca 1 3 = 100.

3x 0 1 7x 2 1

13. Encontre a inversa das seguintes matrizes: 3 5 1 2  0 0   0 0    1 0   2 1  2 2 2 

(a) A =

1 0   2 1  (b) A =   3 2  4 3

  (c) A =  3 4 7    1 2 5   2 0 0   (d) A =  0 3 0    0 0 7

2

14. Classifique e resolva os sistemas: 5x + 8y = 34 10x + 16y = 50   4x − y − 3z = 15   (d) 