Matrizes Vetores e Geometria Analitica
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Matrizes Vetores e Geometria Analitica


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\ud434\ud435 ou \ud435\ud434 e verificar que e´ igual a \ud43c\ud45b.
2.1.9. Se \ud434 e´ uma matriz \ud45b× \ud45b e \ud434\ud458 = 0¯, para \ud458 um inteiro positivo, mostre que
(\ud43c\ud45b \u2212 \ud434)\u22121 = \ud43c\ud45b + \ud434+ \ud4342 + . . .+ \ud434\ud458\u22121 .
Marc¸o 2010 Reginaldo J. Santos
106 Inversa\u2dco de Matrizes e Determinantes
2.1.10. Seja \ud434 uma matriz diagonal, isto e´, os elementos que esta\u2dco fora da diagonal sa\u2dco iguais a zero
(\ud44e\ud456\ud457 = 0, para \ud456 \u2215= \ud457). Se \ud44e\ud456\ud456 \u2215= 0, para \ud456 = 1, . . . , \ud45b, mostre que \ud434 e´ invert\u131´vel e a sua inversa
e´ tambe´m uma matriz diagonal com elementos na diagonal dados por 1/\ud44e11, 1/\ud44e22, . . . , 1/\ud44e\ud45b\ud45b.
2.1.11. Sejam \ud434 e \ud435 matrizes quadradas. Mostre que se \ud434+\ud435 e \ud434 forem invert\u131´veis, enta\u2dco
(\ud434+ \ud435)\u22121 = \ud434\u22121(\ud43c\ud45b + \ud435\ud434\u22121)\u22121.
2.1.12. Seja \ud43d\ud45b a matriz \ud45b× \ud45b, cujas entradas sa\u2dco iguais a 1. Mostre que se \ud45b > 1, enta\u2dco
(\ud43c\ud45b \u2212 \ud43d\ud45b)\u22121 = \ud43c\ud45b \u2212 1
\ud45b\u2212 1\ud43d\ud45b.
(Sugesta\u2dco: observe que \ud43d2\ud45b = \ud45b\ud43d\ud45b.)
2.1.13. Mostre que se \ud435 e´ uma matriz invert\u131´vel, enta\u2dco \ud434\ud435\u22121 = \ud435\u22121\ud434 se, e somente se, \ud434\ud435 = \ud435\ud434.
(Sugesta\u2dco: multiplique a equac¸a\u2dco \ud434\ud435 = \ud435\ud434 por \ud435\u22121.)
2.1.14. Mostre que se \ud434 e´ uma matriz invert\u131´vel, enta\u2dco \ud434 + \ud435 e \ud43c\ud45b + \ud435\ud434\u22121 sa\u2dco ambas invert\u131´veis ou
ambas na\u2dco invert\u131´veis. (Sugesta\u2dco: multiplique \ud434+\ud435 por \ud434\u22121.)
2.1.15. Sejam \ud434 e \ud435 matrizes \ud45b× \ud45b. Mostre que se \ud435 na\u2dco e´ invert\u131´vel, enta\u2dco \ud434\ud435 tambe´m na\u2dco o e´.
2.1.16. Mostre que se \ud434 e \ud435 sa\u2dco matrizes \ud45b× \ud45b, invert\u131´veis, enta\u2dco \ud434 e \ud435 sa\u2dco equivalentes por linhas.
2.1.17. Sejam\ud434 uma matriz\ud45a×\ud45b e\ud435 uma matriz \ud45b×\ud45a, com \ud45b < \ud45a. Mostre que\ud434\ud435 na\u2dco e´ invert\u131´vel.
(Sugesta\u2dco: Mostre que o sistema (\ud434\ud435)\ud44b = 0¯ tem soluc¸a\u2dco na\u2dco trivial.)
Matrizes Vetores e Geometria Anal\u131´tica Marc¸o 2010
2.1 A Inversa de uma Matriz 107
a b c d e f g h i j k l m n
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
o p q r s t u v w x y z a` a´ a\u2c6
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
a\u2dc c¸ e´ e\u2c6 \u131´ o´ o\u2c6 o\u2dc u´ u¨ A B C D E
30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
F G H I J K L M N O P Q R S T
45 46 47 48 49 50 51 52 53 54 55 56 57 58 59
U V W X Y Z A` A´ A\u2c6 A\u2dc C¸ E´ E\u2c6 I´ O´
60 61 62 63 64 65 66 67 68 69 70 71 72 73 74
O\u2c6 O\u2dc U´ U¨ 0 1 2 3 4 5 6 7 8 9 :
75 76 77 78 79 80 81 82 83 84 85 86 87 88 89
; < = > ? @ ! &quot; # $ % & \u2019 ( )
90 91 92 93 94 95 96 97 98 99 100 101 102 103 104
* + , - . / [ \ ] _ { | }
105 106 107 108 109 110 111 112 113 114 115 116 117
Tabela 2.1: Tabela de conversa\u2dco de caracteres em nu´meros
Marc¸o 2010 Reginaldo J. Santos
108 Inversa\u2dco de Matrizes e Determinantes
2.2 Determinantes
Vamos inicialmente definir o determinante de matrizes 1× 1. Para cada matriz \ud434 = [\ud44e] definimos
o determinante de \ud434, indicado por det(\ud434), por det(\ud434) = \ud44e. Vamos, agora, definir o determinante de
matrizes 2×2 e a partir da\u131´ definir para matrizes de ordem maior. A cada matriz \ud434, 2×2, associamos
um nu´mero real, denominado determinante de \ud434, por:
det(\ud434) = det
[
\ud44e11 \ud44e12
\ud44e21 \ud44e22
]
= \ud44e11\ud44e22 \u2212 \ud44e12\ud44e21.
Para definir o determinante de matrizes quadradas maiores, precisamos definir o que sa\u2dco os
menores de uma matriz. Dada uma matriz \ud434 = (\ud44e\ud456\ud457)\ud45b×\ud45b, o menor do elemento \ud44e\ud456\ud457 , denotado por
\ud434\u2dc\ud456\ud457 , e´ a submatriz (\ud45b \u2212 1) × (\ud45b \u2212 1) de \ud434 obtida eliminando-se a \ud456-e´sima linha e a \ud457-e´sima coluna
de \ud434, que tem o seguinte aspecto:
\ud434\u2dc\ud456\ud457 =
\ud457\u23a1
\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a3
\ud44e11 . . .
\u2223\u2223\u2223 . . . \ud44e1\ud45b
.
.
.
\u2223\u2223\u2223\u2223\u2223 ...
\ud44e\ud456\ud457
\u2223\u2223\u2223\u2223\u2223
.
.
.
\u2223\u2223\u2223\u2223\u2223 ...
\ud44e\ud45b1 . . .
\u2223\u2223\u2223 . . . \ud44e\ud45b\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a6
\ud456
Matrizes Vetores e Geometria Anal\u131´tica Marc¸o 2010
2.2 Determinantes 109
Exemplo 2.8. Para uma matriz \ud434 = (\ud44e\ud456\ud457)3×3,
\ud434\u2dc23 =
\u23a1
\u23a2\u23a2\u23a2\u23a3
\ud44e11 \ud44e12 \ud44e13
\u2223\u2223\u2223
\ud44e21 \ud44e22 \ud44e23
\u2223\u2223\u2223
\ud44e31 \ud44e32 \ud44e33
\u2223\u2223\u2223
\u23a4
\u23a5\u23a5\u23a5\u23a6 =
[
\ud44e11 \ud44e12
\ud44e31 \ud44e32
]
Agora, vamos definir os cofatores de uma matriz quadrada \ud434 = (\ud44e\ud456\ud457)3×3. O cofator do elemento
\ud44e\ud456\ud457 , denotado por \ud44e\u2dc\ud456\ud457 , e´ definido por
\ud44e\u2dc\ud456\ud457 = (\u22121)\ud456+\ud457 det(\ud434\u2dc\ud456\ud457),
ou seja, o cofator \ud44e\u2dc\ud456\ud457 , do elemento \ud44e\ud456\ud457 e´ igual a mais ou menos o determinante do menor \ud434\u2dc\ud456\ud457 , sendo
o mais e o menos determinados pela seguinte disposic¸a\u2dco:\u23a1
\u23a3 + \u2212 +\u2212 + \u2212
+ \u2212 +
\u23a4
\u23a6
Exemplo 2.9. Para uma matriz \ud434 = (\ud44e\ud456\ud457)3×3,
\ud44e\u2dc23 = (\u22121)2+3 det(\ud434\u2dc23) = \u2212det
\u23a1
\u23a2\u23a2\u23a2\u23a3
\ud44e11 \ud44e12 \ud44e13
\u2223\u2223\u2223
\ud44e21 \ud44e22 \ud44e23
\u2223\u2223\u2223
\ud44e31 \ud44e32 \ud44e33
\u2223\u2223\u2223
\u23a4
\u23a5\u23a5\u23a5\u23a6 = \u2212det
[
\ud44e11 \ud44e12
\ud44e31 \ud44e32
]
= \ud44e31\ud44e12 \u2212 \ud44e11\ud44e32
Marc¸o 2010 Reginaldo J. Santos
110 Inversa\u2dco de Matrizes e Determinantes
Vamos, agora, definir o determinante de uma matriz 3× 3. Se
\ud434 =
\u23a1
\u23a2\u23a3 \ud44e11 \ud44e12 \ud44e13\ud44e21 \ud44e22 \ud44e23
\ud44e31 \ud44e32 \ud44e33
\u23a4
\u23a5\u23a6 ,
enta\u2dco, o determinante de \ud434 e´ igual a` soma dos produtos dos elementos da 1a. linha pelos seus cofa-
tores.
det(\ud434) = \ud44e11\ud44e\u2dc11 + \ud44e12\ud44e\u2dc12 + \ud44e13\ud44e\u2dc13
= \ud44e11 det
[
\ud44e22 \ud44e23
\ud44e32 \ud44e33
]
\u2212 \ud44e12 det
[
\ud44e21 \ud44e23
\ud44e31 \ud44e33
]
+ \ud44e13 det
[
\ud44e21 \ud44e22
\ud44e31 \ud44e32
]
= \ud44e11(\ud44e22\ud44e33 \u2212 \ud44e32\ud44e23)\u2212 \ud44e12(\ud44e21\ud44e33 \u2212 \ud44e31\ud44e23) + \ud44e13(\ud44e21\ud44e32 \u2212 \ud44e31\ud44e22).
Da mesma forma que a partir do determinante de matrizes 2 × 2, definimos o determinante de
matrizes 3×3, podemos definir o determinante de matrizes quadradas de ordem maior. Supondo que
sabemos como calcular o determinante de matrizes (\ud45b \u2212 1) × (\ud45b \u2212 1) vamos definir o determinante
de matrizes \ud45b× \ud45b.
Vamos definir, agora, os cofatores de uma matriz quadrada \ud434 = (\ud44e\ud456\ud457)\ud45b×\ud45b. O cofator do elemento
\ud44e\ud456\ud457 , denotado por \ud44e\u2dc\ud456\ud457 , e´ definido por
\ud44e\u2dc\ud456\ud457 = (\u22121)\ud456+\ud457 det(\ud434\u2dc\ud456\ud457),
ou seja, o cofator \ud44e\u2dc\ud456\ud457 , do elemento \ud44e\ud456\ud457 e´ igual a mais ou menos o determinante do menor \ud434\u2dc\ud456\ud457 , sendo
Matrizes Vetores e Geometria Anal\u131´tica Marc¸o 2010
2.2 Determinantes 111
o mais e o menos determinados pela seguinte disposic¸a\u2dco:\u23a1
\u23a2\u23a2\u23a2\u23a3
+ \u2212 + \u2212 . . .
\u2212 + \u2212 + . . .
+ \u2212 + \u2212 . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
\u23a4
\u23a5\u23a5\u23a5\u23a6
Definic¸a\u2dco 2.2. Seja \ud434 = (\ud44e\ud456\ud457)\ud45b×\ud45b. O determinante de \ud434, denotado por det(\ud434), e´ definido por
det(\ud434) = \ud44e11\ud44e\u2dc11 + \ud44e12\ud44e\u2dc12 + . . .+ \ud44e1\ud45b\ud44e\u2dc1\ud45b =
\ud45b\u2211
\ud457=1
\ud44e1\ud457 \ud44e\u2dc1\ud457, (2.7)
em que \ud44e\u2dc1\ud457 = (\u22121)1+\ud457 det(\ud434\u2dc1\ud457) e´ o cofator do elemento \ud44e1\ud457 . A expressa\u2dco (2.8) e´ chamada desen-
volvimento em cofatores do determinante de \ud434 em termos da 1a. linha.
Marc¸o 2010 Reginaldo J. Santos
112 Inversa\u2dco de Matrizes e Determinantes
Exemplo 2.10. Seja
\ud434 =
\u23a1
\u23a2\u23a2\u23a2\u23a3
0 0 0 \u22123
1 2 3 4
\u22121 3 2 5
2 1 \u22122 0
\u23a4
\u23a5\u23a5\u23a5\u23a6 .
Desenvolvendo-se o determinante de \ud434 em cofatores, obtemos
det(\ud434) = 0\ud44e\u2dc11 + 0\ud44e\u2dc12 + 0\ud44e\u2dc13 + (\u22123)(\u22121)1+4 det(\ud435), em que \ud435 =
\u23a1
\u23a2\u23a3 1 2 3\u22121 3 2
2 1 \u22122
\u23a4
\u23a5\u23a6 .
Mas o det(\ud435) tambe´m pode ser calculado usando cofatores,
det(\ud435) = 1\ud43511 + 2\ud43512 + 3\ud43513
= 1(\u22121)1+1 det(\ud435\u2dc11) + 2(\u22121)1+2 det(\ud435\u2dc12) + 3(\u22121)1+3 det(\ud435\u2dc13)
= det
[
3 2
1 \u22122
]
\u2212 2 det
[ \u22121 2
2 \u22122
]
+ 3det
[ \u22121 3
2 1
]
= \u22128\u2212 2 (\u22122) + 3 (\u22127)
= \u221225
Portanto,
det(\ud434) = 3 det(\ud435) = \u221275.
Matrizes Vetores e Geometria Anal\u131´tica Marc¸o 2010
2.2 Determinantes 113
Exemplo 2.11. Usando a definic¸a\u2dco de determinante, vamos mostrar que o determinante de uma ma-
triz triangular inferior (isto e´, os elementos situados acima da diagonal principal sa\u2dco iguais a zero) e´
o produto dos elementos da diagonal principal. Vamos mostrar inicialmente para matrizes 3× 3. Seja
\ud434 =
\u23a1
\u23a2\u23a3 \ud44e11 0 0\ud44e21 \ud44e22 0
\ud44e31 \ud44e32 \ud44e33
\u23a4
\u23a5\u23a6
Desenvolvendo-se o determinante de \ud434 em cofatores, obtemos
det(\ud434) = \ud44e11 det
[
\ud44e22 0
\ud44e32 \ud44e33
]
= \ud44e11\ud44e22\ud44e33.
Vamos supor termos provado que para qualquer matriz (\ud45b\u2212 1)× (\ud45b\u2212 1) triangular inferior, o deter-
minante e´ o produto dos elementos da diagonal principal. Enta\u2dco vamos provar que isto tambe´m vale
para matrizes \ud45b× \ud45b. Seja
\ud434 =
\u23a1
\u23a2\u23a2\u23a2\u23a2\u23a3
\ud44e11 0 . . . . . . 0
\ud44e21 \ud44e22 0
.
.
.
.
.
.
.
.
. 0
\ud44e\ud45b1 . . . \ud44e\ud45b\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a5\u23a6
Desenvolvendo-se o determinante de \ud434 em cofatores, obtemos
det(\ud434) = \ud44e11 det
\u23a1
\u23a2\u23a2\u23a2\u23a3
\ud44e22 0 . . . . . . 0
\ud44e32 \ud44e33 0
.
.
.
.
.
.
.
.
. 0
\ud44e\ud45b2 . . . \ud44e\ud45b\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a6 = \ud44e11\ud44e22 . . . \ud44e\ud45b\ud45b,
Marc¸o 2010 Reginaldo J. Santos
114 Inversa\u2dco de Matrizes e Determinantes
pois o determinante acima e´ de uma matriz (\ud45b \u2212 1) × (\ud45b \u2212 1) triangular inferior. Em particular, para
a matriz identidade, \ud43c\ud45b,
det(\ud43c\ud45b) = 1.
2.2.1 Propriedades