# alga.fp1.pdf

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´
Algebra
Linear e Geometria Anal´ıtica
agrupamento IV – ECT, EET, EI
solu¸c˜
oes 1
matrizes e sistemas de equa¸c˜
oes lineares
p´agina 1/3

1.

2.

3.

4.

5.

7.

8.

10.

−4 9
10
 5 −5 −19.
9
39
1

2 0
−1 6
−2 −1 −4
−1

(a) 4 4 ; (b) 1 4 ; (c) 0 −1 0 ; (d)
5
7 9
1 0
3 −2 6

−6 −8 3
3
1 3.
7 −1 6

5
8
.
−2
0
 
2

A primeira coluna ´e 3 e a segunda linha ´e 3 4 .
2

1 2
3
7 2 3
EA = 7 11 15 6= AE = 19 5 6.
7 8
9
31 8 9

 4
µ1 0 · · · 0

.. 
 0 µ42 . . .
. 

.

.
.
.
.
.
.
.
. 0
.
0 ···
0 µ4n

−3
0
; (e)  1
4
8

1 −6
0

1 2 ; (f)
−3
2 16

0
.
−3

i. (A + B)2 = A2 + AB + BA + B 2 ; ii. (A + B)(A − B) = A2 − AB + BA − B 2 ;
iii. (A − B)2 = A2 − AB − BA + B 2 ; iv. (AB)2 = ABAB.

11. (a) Verdadeira; (b) falsa; (c) falsa.

 

−1
1−z
x−y
15. (a) AC =  5 ; (b) AC =
e C = 1 − z , z ∈ R.
2x − y + z
5
z

1 0 0 0
1 0 0

0
3
3
16. ii. e iv. (a) i.
; iii. 0 0 0 5.
0 0 1
0 0 0 0

1 34 0 0

1 0 0 0
1 0 0
0 0 1 0
; iii. 0 0 0 1; iv. 1 0
(b)
i. 0 1 0; ii. 
0 0 0 1
0 1
0 0 0 0
0 0 1
0 0 0 0

11
10

1

3
10
1 .
2

17. (a) x1 = −2, x2 = −10; (b) x1 = 3, x2 = 23 ; (c) x1 = 41 , x2 = 34 + t, x3 = t, t ∈ R; (d) imposs´ıvel;
3
19
20
(e) x1 = t, x2 = 13 − 2t, x3 = t, t ∈ R; (f) x1 = 17
t1 − 13
17 t2 , x2 = 17 t1 − 17 t2 , x3 = t1 , x4 = t2 , t1 , t2 ∈ R;
(g) x1 = 6 − t, x2 = −5 + t, x3 = 3, x4 = −1 − t, x5 = t, t ∈ R; (h) imposs´ıvel.
18. (a) i. α = −1, ii. α 6= 1 e α 6= −1, iii. α = 1; (b) i. α 6= −5, iii. α = −5; (a)
ii. α 6= 1 e α 6= −1, iii. α = −1.

se β = 1;
imposs´ıvel
19. (a) O sistema ´e poss´ıvel e indeterminado de grau um se β = −1;

se β 6= 1 e β 6= −1.
(b) A u
´nica solu¸c˜
ao ´e a solu¸c˜
ao trivial, isto ´e, x = y = z = 0.
20. (a)

i. a ∈ R e b ∈ R \ {−1}; ii. a ∈ R \ {1} e b = −1. (b) {(1, −y, y) : y ∈ R}.

ua

dmat

i. α = 1,