45 pág.

# Matrizes Sistemas

DisciplinaMatemática96.240 materiais2.171.801 seguidores
Pré-visualização10 páginas
```S7
\uf8f4
\uf8f3
\uf8f4
\uf8f2
\uf8f1
\u2212=\u2212
\u2212=\u2212+
\u2212=\u2212+
953
2223
622
zx
zyx
zyx

(c) S3
\uf8f4
\uf8f3
\uf8f4
\uf8f2
\uf8f1
\u2212=++
=++
=++
10435
4453
223
zyx
zyx
zyx
(h) S8
\uf8f4
\uf8f3
\uf8f4
\uf8f2
\uf8f1
=++
=++
=+\u2212
1
643
42
zyx
zyx
zyx

(d) S4
\uf8f4
\uf8f3
\uf8f4
\uf8f2
\uf8f1
\u2212=+\u2212
=++\u2212
=\u2212\u2212
22
2
12
zyx
zyx
zyx
(i) S9
\uf8f3
\uf8f2
\uf8f1
=+\u2212
=\u2212\u2212
2
4
zyx
zyx

(e) S5
\uf8f4
\uf8f3
\uf8f4
\uf8f2
\uf8f1
\u2212=\u2212
\u2212=\u2212
=+
432
123
3
yx
yx
yx
(j) S10
\uf8f4
\uf8f3
\uf8f4
\uf8f2
\uf8f1
=\u2212+
=\u2212+
=\u2212+
3463
2242
032
zyx
zyx
zyx

37. Determine os valores de a e b que tornam o sistema S
\uf8f4
\uf8f4
\uf8f3
\uf8f4
\uf8f4
\uf8f2
\uf8f1
\u2212+=+
+=+
=+
=\u2212
12
2535
73
bayx
bayx
byx
ayx

LISTA DE EXERCÍCIOS

38. Determine o valor de k para que os sistemas abaixo sejam compatíveis indeterminados.

(a) S1
\uf8f4
\uf8f3
\uf8f4
\uf8f2
\uf8f1
=++
=++
=++
073
052
023
zyx
kzyx
zyx
(b) S2
\uf8f4
\uf8f3
\uf8f4
\uf8f2
\uf8f1
\u2212=\u2212
=\u2212
=\u2212
kxy
zx
yz
332
224
143
(c) S3
\uf8f4
\uf8f3
\uf8f4
\uf8f2
\uf8f1
=\u2212+\u2212
=+\u2212
\u2212=\u2212+
kzyx
zyx
z,yx
242
43
1602

39. Discuta em função de k os seguintes sistemas:

Cursos de Engenharia
Disciplinas: Geometria Analítica e Álgebra Linear
Profa. Andréa Cirino e Prof. Clevenson Atanásio

40
(a) S1
\uf8f4
\uf8f3
\uf8f4
\uf8f2
\uf8f1
=\u2212
=\u2212
=+\u2212
kyx
yx
yx
2
045
234
(c) S3
\uf8f4
\uf8f3
\uf8f4
\uf8f2
\uf8f1
=+\u2212
=+\u2212
=+\u2212
0
32
222
zkyx
kzyx
kzyx

(b) S2
\uf8f3
\uf8f2
\uf8f1
=\u2212+
=\u2212+
2
0
zykx
kzyx
(d) S4
\uf8f4
\uf8f3
\uf8f4
\uf8f2
\uf8f1
\u2212=++
=\u2212\u2212
\u2212=+
54
2
2
zkyx
kzyx
kzx

RESPOSTAS

1. \uf8f7\uf8f7
\uf8f8
\uf8f6
\uf8ec\uf8ec
\uf8ed
\uf8eb \u2212
21
41
2. 10

3. ( )
\uf8fa
\uf8fa
\uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8ef
\uf8ef
\uf8f0
\uf8ee
=\u22c5
1212
4634
4043
tBA 4.
\uf8f7
\uf8f7
\uf8f7
\uf8f8
\uf8f6
\uf8ec
\uf8ec
\uf8ec
\uf8ed
\uf8eb
\u2212\u2212\u2212
\u2212\u2212
\u2212
=
366
624
641
C é uma matriz anti-simétrica

5. (a) 5 \u22c5 A \u2013 2 \u22c5 B =

3427
105
\uf8f7\uf8f7
\uf8f8
\uf8f6
\uf8ec\uf8ec
\uf8ed
\uf8eb
\u2212
\u2212
e 2 \u22c5 A + 3 \u22c5 B = \uf8f7\uf8f7
\uf8f8
\uf8f6
\uf8ec\uf8ec
\uf8ed
\uf8eb
\u2212 1312
417

(b) A2 =

229
67
\uf8f7\uf8f7
\uf8f8
\uf8f6
\uf8ec\uf8ec
\uf8ed
\uf8eb
\u2212
\u2212
e A \u22c5 C = \uf8f7\uf8f7
\uf8f8
\uf8f6
\uf8ec\uf8ec
\uf8ed
\uf8eb
\u2212\u2212
\u2212
32335
695

6. (a) Ryx
x
yx
\u2208\u2200\uf8f7\uf8f7
\uf8f8
\uf8f6
\uf8ec\uf8ec
\uf8ed
\uf8eb
, ,
0
(b) Rzyx
x
yx
zyx
\u2208\u2200
\uf8f7
\uf8f7
\uf8f7
\uf8f8
\uf8f6
\uf8ec
\uf8ec
\uf8ec
\uf8ed
\uf8eb
,, ,
00
0

7. (a) 0=x (b) 2\u2212=x

8.
\uf8f7
\uf8f7
\uf8f7
\uf8f8
\uf8f6
\uf8ec
\uf8ec
\uf8ec
\uf8ed
\uf8eb
\u2212
\u2212=
930
366
068
S

9. \uf8f7\uf8f7
\uf8f8
\uf8f6
\uf8ec\uf8ec
\uf8ed
\uf8eb
\u2212\u2212\u2212
=
321
642
B . Existem outras.
12. ( ) OAg =

13. (a) \uf8f7\uf8f7
\uf8f8
\uf8f6
\uf8ec\uf8ec
\uf8ed
\uf8eb
\u2212
=
1
1
X (b)
\uf8f7
\uf8f7
\uf8f7
\uf8f8
\uf8f6
\uf8ec
\uf8ec
\uf8ec
\uf8ed
\uf8eb
\u2212=
1
3
5
Y (c) \uf8f7\uf8f7
\uf8f8
\uf8f6
\uf8ec\uf8ec
\uf8ed
\uf8eb \u2212\u2212
=
130
211
W

RESPOSTAS

14. (a) 10 (b) ( )yx +cos (c) ( )sen x y+
(d) 1 (e) 49 (f) \u20136
(g) 48 (h) a2 + b2 (i) abcd

Cursos de Engenharia
Disciplinas: Geometria Analítica e Álgebra Linear
Profa. Andréa Cirino e Prof. Clevenson Atanásio

41
15. (a) 21ou 1 =\u2212= xx (b) 1ou 0 == xx 16. 25

17. (a) 11 .. \u2212\u2212= BCAX (b) BIX \u2212= (c) ( ) 11 ... \u2212\u2212= CACBX

(d) BX = (e) ( ) BAABX tt ... 11 \u2212\u2212=

18. 1\u2212=\u21d2=\u22c5 MMIMM tt . M é chamada de matriz ortogonal.

19. (a) \uf8f7\uf8f7
\uf8f8
\uf8f6
\uf8ec\uf8ec
\uf8ed
\uf8eb
\u2212
\u2212
=
\u2212
12
371A (b)
\uf8f7
\uf8f7
\uf8f7
\uf8f8
\uf8f6
\uf8ec
\uf8ec
\uf8ec
\uf8ed
\uf8eb
\u2212
\u2212
\u2212
=
\u2212
2711274278
274271272
616161
1B

(c) C não é inversível.

20. P(A) = 2 e N(A) = 1 P(B) = 2 = N(B) P(C) = 2 e N(C) = 0
P(D) = 2 e N(D) = 0 P(E) = 3 e N(E) = 0

21. (a) B = \uf8f7\uf8f7
\uf8f8
\uf8f6
\uf8ec\uf8ec
\uf8ed
\uf8eb
010
001
(b) Impossível (c) Impossível

(d) F = \uf8f7\uf8f7
\uf8f8
\uf8f6
\uf8ec\uf8ec
\uf8ed
\uf8eb
000
001
(e) G =
\uf8f7\uf8f7
\uf8f7
\uf8f7
\uf8f7
\uf8f8
\uf8f6
\uf8ec\uf8ec
\uf8ec
\uf8ec
\uf8ec
\uf8ed
\uf8eb
000
100
010
001
(f) H =
\uf8f7
\uf8f7
\uf8f7
\uf8f8
\uf8f6
\uf8ec
\uf8ec
\uf8ec
\uf8ed
\uf8eb
100
010
001

(g) J =
\uf8f7
\uf8f7
\uf8f7
\uf8f8
\uf8f6
\uf8ec
\uf8ec
\uf8ec
\uf8ed
\uf8eb
000
010
001
OBSERVAÇÃO. Estes exemplos não são únicos.
22. a + b + c = \u20134 23. det A = 0

24. (A) V, (B) V, (C) V, (D) V, (E) V, (F) F, (G) F, (H) F, (I) F, (J) V, (K) F, (L) V, (M) F

25. (a)
Matriz dos coeficientes: \uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8f0
\uf8ee \u2212\u2212
23
41
Matriz das incógnitas: \uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8f0
\uf8ee
y
x

Matriz dos termos independentes: \uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8f0
\uf8ee
5
0
\uf8fb
\uf8f9
\uf8ef
\uf8f0
\uf8ee \u2212\u2212
523
041

(b)
Matriz dos coeficientes: \uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8f0
\uf8ee
\u2212
\u2212
31
12
Matriz das incógnitas: \uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8f0
\uf8ee
y
x

Matriz dos termos independentes: \uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8f0
\uf8ee
\u2212 3
2
\uf8fb
\uf8f9
\uf8ef
\uf8f0
\uf8ee
\u2212\u2212
\u2212
331
212

RESPOSTAS

Cursos de Engenharia
Disciplinas: Geometria Analítica e Álgebra Linear
Profa. Andréa Cirino e Prof. Clevenson Atanásio

42

(c)
Matriz dos coeficientes:
\uf8fa
\uf8fa
\uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8ef
\uf8ef
\uf8f0
\uf8ee
\u2212
\u2212
214
302
113
Matriz das incógnitas:
\uf8fa
\uf8fa
\uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8ef
\uf8ef
\uf8f0
\uf8ee
z
y
x

Matriz dos termos independentes:
\uf8fa
\uf8fa
\uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8ef
\uf8ef
\uf8f0
\uf8ee
\u2212
7
1
1
\uf8fa
\uf8fa
\uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8ef
\uf8ef
\uf8f0
\uf8ee
\u2212
\u2212
\u2212
7214
1302
1113

(d)
Matriz dos coeficientes:
\uf8fa
\uf8fa
\uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8ef
\uf8ef
\uf8f0
\uf8ee
\u2212
\u2212\u2212
231
412
111
Matriz das incógnitas:
\uf8fa
\uf8fa
\uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8ef
\uf8ef
\uf8f0
\uf8ee
z
y
x

Matriz dos termos independentes:
\uf8fa
\uf8fa
\uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8ef
\uf8ef
\uf8f0
\uf8ee
\u2212 3
4
5
\uf8fa
\uf8fa
\uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8ef
\uf8ef
\uf8f0
\uf8ee
\u2212\u2212
\u2212\u2212
3231
4412
5111

(e)
Matriz dos coeficientes:
\uf8fa
\uf8fa
\uf8fa
\uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8ef
\uf8ef
\uf8ef
\uf8f0
\uf8ee
\u2212\u2212\u2212
\u2212
1202
1111
0112
1111
Matriz das incógnitas:
\uf8fa
\uf8fa
\uf8fa
\uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8ef
\uf8ef
\uf8ef
\uf8f0
\uf8ee
t
z
y
x

Matriz dos termos independentes:
\uf8fa
\uf8fa
\uf8fa
\uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8ef
\uf8ef
\uf8ef
\uf8f0
\uf8ee
\u22121
0
2
1
\uf8fa
\uf8fa
\uf8fa
\uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8ef
\uf8ef
\uf8ef
\uf8f0
\uf8ee
\u2212
\u2212\u2212\u2212
\u2212
11202
01111
20112
11111

(f)
Matriz dos coeficientes:
\uf8fa
\uf8fa
\uf8fa
\uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8ef
\uf8ef
\uf8ef
\uf8f0
\uf8ee
\u2212
\u2212\u2212\u2212
\u2212
2131
1112
0121
1111
Matriz das incógnitas:
\uf8fa
\uf8fa
\uf8fa
\uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8ef
\uf8ef
\uf8ef
\uf8f0
\uf8ee
t
z
y
x

Matriz dos termos independentes:
\uf8fa
\uf8fa
\uf8fa
\uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8ef
\uf8ef
\uf8ef
\uf8f0
\uf8ee
\u2212
0
1
2
1
\uf8fa
\uf8fa
\uf8fa
\uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8ef
\uf8ef
\uf8ef
\uf8f0
\uf8ee
\u2212
\u2212\u2212\u2212\u2212
\u2212
02131
11112
20121
11111

26. (a) \uf8f7
\uf8f8
\uf8f6
\uf8ec
\uf8ed
\uf8eb
\u2212
2
1
,2 (b) \uf8f7
\uf8f8
\uf8f6
\uf8ec
\uf8ed
\uf8eb
\u2212
5
4
,
5
3
(c) ( )1,1,1 \u2212

(d) (\u20132,3,0) (e) \uf8f7
\uf8f8
\uf8f6
\uf8ec
\uf8ed
\uf8eb
\u2212 2
2
11
2
14 ,,, (f) (0,0,2,\u20131)

27. Todos os sistemas são compatíveis determinados (têm solução única), pois o determinante da matriz
dos coeficientes de cada sistema é diferente de zero.

Cursos de Engenharia
Disciplinas: Geometria Analítica e Álgebra Linear
Profa. Andréa Cirino e Prof. Clevenson Atanásio

43
28. Está na forma escalonada o sistema dos itens (a), (b) e (e).

RESPOSTAS

29.
LRFEA =
\uf8f7
\uf8f7
\uf8f7
\uf8f7
\uf8f7
\uf8f7
\uf8f8
\uf8f6
\uf8ec
\uf8ec
\uf8ec
\uf8ec
\uf8ec
\uf8ec
\uf8ed
\uf8eb
\u2212
1000
0
6
110
0
3
201
LRFEB =
\uf8f7
\uf8f7
\uf8f7
\uf8f8
\uf8f6
\uf8ec
\uf8ec
\uf8ec
\uf8ed
\uf8eb
000
010
001

LRFEC =
\uf8f7
\uf8f7
\uf8f7
\uf8f7
\uf8f8
\uf8f6
\uf8ec
\uf8ec
\uf8ec
\uf8ec
\uf8ed
\uf8eb
\u2212
\u2212
0
5
210
1
5
401

LRFED =
\uf8f7
\uf8f7
\uf8f7
\uf8f8
\uf8f6
\uf8ec
\uf8ec
\uf8ec
\uf8ed
\uf8eb```