Gabarito dos Exercícios de Álgebra Lienar

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UNIVERSIDADE FEDERAL DO RIO GRANDE
DISCIPLINA: A´LGEBRA LINEAR
GABARITO
Exerc´ıcios: Sec¸a˜o 1.4.1
questa˜o 3 pa´gina 9
(%i1) A:matrix([6,5],[-9,7]);
(%o1)
(
6 5
−9 7
)
(%i2) B:matrix([5,8],[10,7]);
(%o2)
(
5 8
10 7
)
(%i3) C:matrix([10,-3,4],[5,7,4],[-9,1,4]);
(%o3)
10 −3 45 7 4
−9 1 4

(%i4) D:matrix([5,1,-2],[3,6,7]);
(%o4)
(
5 1 −2
3 6 7
)
(%i5) r3a:A.B-B.A;
(%o5)
(
122 2
28 −122
)
(%i6) r3b:4*A+D;
fullmap : argumentsmusthavesameformalstructure.−−anerror.Todebugthistry :
debugmode(true);
(%i7) r3c:2*C;
(%o7)
 20 −6 810 14 8
−18 2 8

(%i8) r3d:0.5*D;
(%o8)
(
2.5 0.5 −1.0
1.5 3.0 3.5
)
questa˜o 4 pa´gina 9
1
(%i9) A4:matrix([4,0,3],[1/2,2,1],[-4,8,-9]);
(%o9)
 4 0 31
2 2 1−4 8 −9

(%i10) B4:matrix([1,-3,2],[2,5,-3],[4,2,0]);
(%o10)
1 −3 22 5 −3
4 2 0

(%i11) C4:matrix([1,5],[0,2],[4,3]);
(%o11)
1 50 2
4 3

(%i12) r4a:transpose(A4.B4);
(%o12)
16 172 −24−6 212 34
8 −5 −32

(%i13) r4b:transpose(4*C4);
(%o13)
(
4 0 16
20 8 12
)
questa˜o 5 pa´gina 9
(%i14) A5:matrix([4,10,1],[0,5,-9],[3,7,2]);
(%o14)
4 10 10 5 −9
3 7 2

(%i15) B5:matrix([0,-5,2],[7,1,7],[3,2,1]);
(%o15)
0 −5 27 1 7
3 2 1

(%i16) C5:matrix([2,1,4],[-3,2,-2],[9,1,-2]);
(%o16)
 2 1 4−3 2 −2
9 1 −2

(%i17) r5a:A5+transpose(A5);
2
(%o17)
 8 10 410 10 −2
4 −2 4

(%i18) r5b:A5.transpose(A5);
(%o18)
117 41 8441 106 17
84 17 62

(%i19) r5c:B5-transpose(B5);
(%o19)
 0 −12 −112 0 5
1 −5 0

(%i20) r5c:C5-transpose(C5);
(%o20)
 0 4 −5−4 0 −3
5 3 0

Exerc´ıcios: Sec¸a˜o 1.6.1 pa´gina 13
Exerc´ıcio 1
(%i21) determinant(matrix([4,6,x],[5,2,-x],[7,4,2*x]))=-128;
(%o21) − 64x = −128
Portanto, x=2.
Exerc´ıcio 2
(%i22) determinant(matrix([x+3,x+1,x+4],[4,5,3],[9,10,7]))=-7;
(%o22) − 5 (x+ 4) + 5 (x+ 3)− x− 1 = −7
Portanto, x=1.
Exerc´ıcio 3
(%i23) M1:determinant(matrix([3,0,6],[3,0,2],[4,-7,8]));
(%o23) − 84
Exerc´ıcio 4
(%i24) M2:matrix([3,6,4,0],[3,2,0,1],[0,2,-1,2],[2,0,2,0]);
(%o24)

3 6 4 0
3 2 0 1
0 2 −1 2
2 0 2 0

3
(%i25) determinant(M2);
(%o25) 64
Exerc´ıcio 6
(%i26) M6a:determinant(matrix([2,4],[4,6]));
(%o26) − 4
(%i27) M6b:determinant(matrix([4,3],[1,4]));
(%o27) 13
Exerc´ıcio 7
(%i28) M5:5*(matrix([3,1],[4,2]));
(%o28)
(
15 5
20 10
)
(%i29) M5a:determinant(M5);
(%o29) 50
Lista 1 pg 14 Exerc´ıcio 2
(%i30) L2:linsolve([x/2=x+2,2*y=9+x+y,-2*z=6+z,2*t=5+z], [x,y,z,t]);
(%o30) [x = −4, y = 5, z = −2, t = 3
2
]
Exerc´ıcio 3
(%i31) L3a:3*matrix([3,6],[4,4])+matrix([-7,3],[3,-1]);
(%o31)
(
2 21
15 11
)
(%i32) L3b:3*matrix([3,6],[4,4])-2*matrix([9,-8],[8,5]);
(%o32)
(−9 34
−4 2
)
(%i33) L3c:matrix([9,-8],[8,5]).matrix([9,-8],[8,5]);
(%o33)
(
17 −112
112 −39
)
(%i40) L3d:matrix([-7,3],[3,-1])^^3;
(%o40)
(−478 198
198 −82
)
4
Exerc´ıcio 4:
(%i44) L4:matrix([-4,-1,-4],[0,-3,6]).matrix([3,3,7,2],[2,9,-4,4],[5,3,3,0]);
(%o44)
(−34 −33 −36 −12
24 −9 30 −12
)
Exerc´ıcio 5:
(%i45) L5:transpose(matrix([2,6,-3],[12,-6,7]));
(%o45)
 2 126 −6
−3 7

(%i46) L5:transpose(matrix([1,2,3],[2,3,7],[1,6,4]));
(%o46)
1 2 12 3 6
3 7 4

(%i47) L5:transpose(matrix([2,-7],[10,2],[4,5]));
(%o47)
(
2 10 4
−7 2 5
)
Exerc´ıcio 6: Matriz A(2x3) B(3x4) = C(2x4) Exerc´ıcio 7: C precisa ter 3 linhas
Exerc´ıcio 8:
(%i48) L8:matrix([3,-2],[5,3])^^2;
(%o48)
(−1 −12
30 −1
)
(%i49) L8:matrix([3,-2],[5,3])^^3;
(%o49)
(−63 −34
85 −63
)
(%i50) L8:matrix([3,-2],[5,3])^^2+2*matrix([3,-2],[5,3])+6*matrix([1,0],[0,1]);
(%o50)
(
11 −16
40 11
)
Exerc´ıcio 9: alpha=8 e beta=-12.
Exerc´ıcio 10: a. y=5; z=4; x=6 b. y=4; x=0; z qualquer valor.
Exerc´ıcio 11:
(%i51) L11:matrix([2,2,0],[4,4,2]).matrix([600],[750],[420]);
(%o51)
(
2700
6240
)
5
Exerc´ıcio 12:
(%i52) L12:matrix([2,1,2],[3,4,2]).matrix([60,100],[40,40],[50,50]);
(%o52)
(
260 340
440 560
)
Exerc´ıcio 13:
(%i53) L13:matrix([5,4],[8,6],[10,8]).matrix([10,8],[14,6]);
(%o53)
106 64164 100
212 128

Exerc´ıcio 14:
(%i55) L14a:matrix([6,5,10]).matrix([10,14,12,6,17],[8,12,16,5,10],[7,4,10,3,12]);
(%o55)
(
170 184 252 91 272
)
(%i58) L14b:matrix([10,14,12,6,17],[8,12,16,5,10],[7,4,10,3,12]).matrix([6],[9],[4],[1],[5]);
(%o58)
325275
181

Exerc´ıcio 15:
(%i59) L15:matrix([2,3]).matrix([4,1,0],[1,4,2]);
(%o59)
(
11 14 6
)
Exerc´ıcio 16:
(%i60) L15:matrix([50,70]).matrix([10,12,4],[7,5,2]);
(%o60)
(
990 950 340
)
(%i61) L15b:matrix([30,40]).matrix([10,12,4],[7,5,2]);
(%o61)
(
580 560 200
)
Sistemas Lineares Sec¸a˜o 1.8.2 pa´gina 29 Exerc´ıcio 1:
(%i62) S1:linsolve([4*x+2*y+z=4, 16*x-4*y+z=4, x+y+z=-1], [x,y,z]);
(%o62) [x = 1, y = 2, z = −4]
Exerc´ıcio 2:
6
(%i64) S2:linsolve([3*a+b+c+10=0, 6*a-2*b+c+40=0, 3*a-5*b+c+34=0], [a,b,c]);
(%o64) [a = −6, b = 4, c = 4]
Exerc´ıcio 3:
(%i67) S3:linsolve([-a-b+c+2=0, 3*a-5*b+c+24=0, 7*a-b+c+50=0], [a,b,c]);
(%o67) [a = −6, b = −1
2
, c = −17
2
]
Exerc´ıcio 4: Resolver o sistema e analisar as poss´ıveis soluc¸o˜es dependendo dos
valores de α e β
Exerc´ıcio pg 30
(%i63) linsolve([3*a+b+c+10=0, 6*a-2*b+c+40=0, 3*a-5*b+c+34=0], [a,b,c]);
(%o63) [a = −6, b = 4, c = 4]
Exerc´ıcio 5:
(%i70) S5:linsolve([4*x1+5*x2+2*x3=42, 3*x1+2*x2+2*x3=27, 2*x1+3*x2+3*x3=33], [x1,x2,x3]);
(%o70) [x1 = 3, x2 = 4, x3 = 5]
Exerc´ıcio 6:
(%i71) S6:linsolve([c+e=60, 4*c-2*e=180], [c,e]);
(%o71) [c = 50, e = 10]
Sec¸a˜o 1.8.3 pa´gina 30
(%i72) invert(matrix([2,2,1],[4,1,2],[3,1,1]));
(%o72)
− 13 − 13 12
3 − 13 0
1
3
4
3 −2

(%i73) invert(matrix([-1,0,1],[0,-2,3],[3,0,1]));
(%o73)
− 14 0 149
8 − 12 38
3
4 0
1
4

Sec¸a˜o 1.8.4 pa´gina 31
(%i74) linsolve([2*x+4*y=16, 5*x-2*y=4, 3*x+y=9,4*x-5*y=-7], [x,y]);
solve : dependentequationseliminated : (43)
(%o74) [x = 2, y = 3]
7
(%i75) linsolve([2*x-8*y+24*z+18*w=84, 4*x-14*y+52*z+42*w=190], [x,y,z,w]);
(%o75) [x = −20 %r2−21 %r1+86, y = −2 %r2−3 %r1+11, z = %r2, w = %r1]
Lista 2 pa´gina 33
Exerc´ıcio 1
(%i76) L21:linsolve([2*x+4*y+3*z=-4, 4*y+5*z=2, -2*z=0], [x,y,z]);
(%o76) [x = −3, y = 1
2
, z = 0]
Exerc´ıcio 3
(%i77) L23a:linsolve([2*y+10*z=-4, 2*x+8*y+6*z=-2, 4*x+14*y+2*z=-1], [x,y,z]);
(%o77) []
(%i83) L23b:linsolve([x+3*y+4*z=8, 2*x-y+1/2*z=4, 3*x-1/2*y+1/2*z=5], [x,y,z]);
(%o83) [x =
46
37
, y = −18
37
, z =
76
37
]
(%i84) L23c:linsolve([2*x-3*y+4*z=8, 2*x+8*y+13*z=23, 1/2*x+y+2*z=10], [x,y,z]);
(%o84) [x = 48, y = 12, z = −13]
(%i85) L23d:linsolve([2*x+4*y+6*z=12, x-z=0, 5/2*x+2*y+11/2*z=12], [x,y,z]);
(%o85) [x =
3
2
, y = 0, z =
3
2
]
(%i86) L23e:linsolve([2*x-12*y=5, 2*y-8*z+2*w=0, -2*x+12*y+2*z+10*w=3,-2*y+10*z+8*w=0], [x,y,z,w]);
(%o86) []
(%i87) L23f:linsolve([-x+2*z=5, y+3*z=2, 2*x+4*y+7*z=-5], [x,y,z]);
(%o87) [x = 1, y = −7, z = 3]
Exerc´ıcio 4
(%i88) L24:invert(matrix([0,1,1],[0,1,2],[1,1,2]));
(%o88)
 0 −1 12 −1 0
−1 1 0

Exerc´ıcio 5:
8
(%i89) L25:linsolve([2*a-b+c=-5, -3*a+c=-9, a+4*b+c=-17], [a,b,c]);
(%o89) [a =
1
3
, b = −7
3
, c = −8]
Exerc´ıcio 6:
(%i90) L26:invert(matrix([0,1,2],[1,0,3],[2,-3,4]));
(%o90)
− 94 52 − 34− 12 1 − 12
3
4 − 12 14

Exerc´ıcio 8:
(%i92) L28a:invert(matrix([2,-6,4],[4,16,-6],[-1,2,1]));
(%o92)
 15 110 − 151
70
3
70
1
5
6
35
1
70
2
5

(%i93) L28b:invert(matrix([2,1/2,2],[1,-4,-1],[-1,-2,-2]));
(%o93)
 4 −2 52 − 43 83−4 73 − 173

Exerc´ıcio 9
(%i94) L29a:linsolve([x+2*y-z=0, x+y-z=0, 2*x-2*y-z=0], [x,y,z]);
(%o94) [x = 0, y = 0, z = 0]
(%i95)L29b:linsolve([x+2*y+3*z=0, x+y+z=0, x+y+2*z=0,x+3*y+3*z=0], [x,y,z]);
solve : dependentequationseliminated : (2)
(%o95) [x = 0, y = 0, z = 0]
(%i96) L29c:linsolve([x-2*y+4*z=0, 2*x+5*y-3*z=0, 3*x-y+2*z=0], [x,y,z]);
(%o96) [x = 0, y = 0, z = 0]
(%i97) L29d:linsolve([2*x+2*y=0, 3*x+5*y=0, 4*x+3*y+3*z=0], [x,y,z]);
(%o97) [x = 0, y = 0, z = 0]
Exerc´ıcio 10:
(%i98) L210:linsolve([x+2*y-z=3, 2*x+3*y+z=1], [x,y]);
(%o98) [x = −5 z − 7, y = 3 z + 5]
9
Exerc´ıcio 11:
(%i99) L211:linsolve([x+3*z=0, 3*y-2*w=0,y-2*z=0,4*y-8*z=0], [x,y,z,w]);
solve : dependentequationseliminated : (4)
(%o99) [x = −3 %r3
2
, y = %r3, z =
%r3
2
, w =
3 %r3
2
]
Exerc´ıcio 12:
(%i100)L212:linsolve([3*x-z=0, 8*x-2*w=0,2*y-2*z-w=0], [x,y,z,w]);
(%o100)[x =
%r4
4
, y =
5 %r4
4
, z =
3 %r4
4
, w = %r4]
Exerc´ıcio 13:
(%i101)L213:linsolve([2*x-z=0, 6*x-2*w=0,x+2*y-2*z-w=0], [x,y,z,w]);
(%o101)[x =
%r5
3
, y = %r5, z =
2 %r5
3
, w = %r5]
Exerc´ıcio 14:
(%i102)L214:linsolve([4*c1+4*c2=36,2*c1+c2=16], [c1,c2]);
(%o102)[c1 = 7, c2 = 2]
Exerc´ıcio 15:
(%i103)L215:linsolve([2*x+5*y+2*z=140, 2*x+2*y+2*z=98,2*x+3*y+4*z=160], [x,y,z]);
(%o103)[x = 11, y = 14, z = 24]
Exerc´ıcio 16:
(%i104)L216:linsolve([25*x+300*y+100*z=1350, 2*x+12*y+14*z=66,0.10*x+0.30*y+0.40*z=1.8], [x,y,z]);
rat : replaced−1.8by−9/5 = −1.8rat : replaced0.1by1/10 = 0.1rat : replaced0.3by3/10 =
0.3rat : replaced0.4by2/5 = 0.4
(%o104)[x = 2, y = 4, z = 1]
10