Matrizes Sistemas
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Matrizes Sistemas


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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
, possível determinado. 
 
 
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: 
 
 
 
UNIVERSIDADE SALVADOR (UNIFACS ) 
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 
 
 
 
UNIVERSIDADE SALVADOR (UNIFACS ) 
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
 Matriz ampliada: \uf8fa
\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
 Matriz ampliada: \uf8fa
\uf8fb
\uf8f9
\uf8ef
\uf8f0
\uf8ee
\u2212\u2212
\u2212
331
212
 
 
 
 
 
 
RESPOSTAS 
 
 
UNIVERSIDADE SALVADOR (UNIFACS ) 
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
 Matriz ampliada: 
\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
 Matriz ampliada: 
\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
 Matriz ampliada: 
\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
 Matriz ampliada: 
\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. 
 
 
 
UNIVERSIDADE SALVADOR (UNIFACS ) 
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