Matrizes  Vetores e Geometria Analítica
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Matrizes Vetores e Geometria Analítica


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\ud434\ud461 tambe´m na\u2dco o e´, pois caso contra´rio, pelo Teorema 2.2 na
pa´gina 81, tambe´m \ud434 = (\ud434\ud461)\ud461 seria invert\u131´vel. Assim neste caso, det(\ud434\ud461) = 0 = det(\ud434). \u25a0
2.2.3 Matriz Adjunta e Inversa\u2dco (opcional)
Vamos definir a adjunta de uma matriz quadrada e em seguida enunciar e provar um teorema
sobre a adjunta que permite provar va´rios resultados sobre matrizes, entre eles um que fornece uma
fo´rmula para a inversa de uma matriz e tambe´m a regra de Cramer. Tanto a adjunta quanto os
resultados que vem a seguir sa\u2dco de importa\u2c6ncia teo´rica.
Definic¸a\u2dco 2.3. Seja \ud434 uma matriz \ud45b× \ud45b. Definimos a matriz adjunta (cla´ssica) de \ud434, denotada por
adj(\ud434), como a transposta da matriz formada pelos cofatores de \ud434, ou seja,
adj(\ud434) =
\u23a1
\u23a2\u23a2\u23a2\u23a3
\ud44e\u2dc11 \ud44e\u2dc12 . . . \ud44e\u2dc1\ud45b
\ud44e\u2dc21 \ud44e\u2dc22 . . . \ud44e\u2dc2\ud45b
.
.
. . . .
.
.
.
\ud44e\u2dc\ud45b1 \ud44e\u2dc\ud45b2 . . . \ud44e\u2dc\ud45b\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a6
\ud461
=
\u23a1
\u23a2\u23a2\u23a2\u23a3
\ud44e\u2dc11 \ud44e\u2dc21 . . . \ud44e\u2dc\ud45b1
\ud44e\u2dc12 \ud44e\u2dc22 . . . \ud44e\u2dc\ud45b2
.
.
. . . .
.
.
.
\ud44e\u2dc1\ud45b \ud44e\u2dc2\ud45b . . . \ud44e\u2dc\ud45b\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a6 ,
em que, \ud44e\u2dc\ud456\ud457 = (\u22121)\ud456+\ud457 det(\ud434\u2dc\ud456\ud457) e´ o cofator do elemento \ud44e\ud456\ud457 , para \ud456, \ud457 = 1, . . . , \ud45b.
Matrizes Vetores e Geometria Anal\u131´tica Marc¸o 2010
2.2 Determinantes 131
Marc¸o 2010 Reginaldo J. Santos
132 Inversa\u2dco de Matrizes e Determinantes
Exemplo 2.19. Seja
\ud435 =
\u23a1
\u23a3 1 2 30 3 2
0 0 \u22122
\u23a4
\u23a6 .
Vamos calcular a adjunta de \ud435.
\ud44f\u2dc11 = (\u22121)1+1 det
[
3 2
0 \u22122
]
= \u22126, \ud44f\u2dc12 = (\u22121)1+2 det
[
0 2
0 \u22122
]
= 0,
\ud44f\u2dc13 = (\u22121)1+3 det
[
0 3
0 0
]
= 0, \ud44f\u2dc21 = (\u22121)2+1 det
[
2 3
0 \u22122
]
= 4,
\ud44f\u2dc22 = (\u22121)2+2 det
[
1 3
0 \u22122
]
= \u22122, \ud44f\u2dc23 = (\u22121)2+3 det
[
1 2
0 0
]
= 0,
\ud44f\u2dc31 = (\u22121)3+1 det
[
2 3
3 2
]
= \u22125, \ud44f\u2dc32 = (\u22121)3+2 det
[
1 3
0 2
]
= \u22122,
\ud44f\u2dc33 = (\u22121)3+3 det
[
1 2
0 3
]
= 3,
Assim, a adjunta de \ud435 e´
adj(\ud435) =
\u23a1
\u23a3 \u22126 0 04 \u22122 0
\u22125 \u22122 3
\u23a4
\u23a6\ud461 =
\u23a1
\u23a3 \u22126 4 \u221250 \u22122 \u22122
0 0 3
\u23a4
\u23a6
Na definic¸a\u2dco do determinante sa\u2dco multiplicados os elementos de uma linha pelos cofatores da
mesma linha. O teorema seguinte diz o que acontece se somamos os produtos dos elementos de
uma linha com os cofatores de outra linha ou se somamos os produtos dos elementos de uma coluna
com os cofatores de outra coluna.
Matrizes Vetores e Geometria Anal\u131´tica Marc¸o 2010
2.2 Determinantes 133
Lema 2.17. Se \ud434 e´ uma matriz \ud45b× \ud45b, enta\u2dco
\ud44e\ud4581\ud44e\u2dc\ud4561 + \ud44e\ud4582\ud44e\u2dc\ud4562 + . . .+ \ud44e\ud458\ud45b\ud44e\u2dc\ud456\ud45b = 0 se \ud458 \u2215= \ud456; (2.10)
\ud44e1\ud458\ud44e\u2dc1\ud457 + \ud44e2\ud458\ud44e\u2dc2\ud457 + . . .+ \ud44e\ud45b\ud458\ud44e\u2dc\ud45b\ud457 = 0 se \ud458 \u2215= \ud457; (2.11)
em que, \ud44e\u2dc\ud456\ud457 = (\u22121)\ud456+\ud457 det(\ud434\u2dc\ud456\ud457) e´ o cofator do elemento \ud44e\ud456\ud457 , para \ud456, \ud457 = 1, . . . , \ud45b.
Demonstrac¸a\u2dco. Para demonstrar a equac¸a\u2dco (2.10), definimos a matriz \ud434\u2217 como sendo a matriz
obtida de \ud434 substituindo a \ud456-e´sima linha de \ud434 por sua \ud458-e´sima linha, ou seja,
\ud434 =
\u23a1
\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a3
\ud4341
.
.
.
\ud434\ud456
.
.
.
\ud434\ud458
.
.
.
\ud434\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a6
\u2190 \ud456
\u2190\ud458
e \ud434\u2217 =
\u23a1
\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a3
\ud4341
.
.
.
\ud434\ud458
.
.
.
\ud434\ud458
.
.
.
\ud434\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a6
\u2190 \ud456
\u2190\ud458
.
Assim, \ud434\u2217 possui duas linhas iguais e pelo Corola´rio 2.12 na pa´gina 118, det(\ud434\u2217) = 0. Mas, o
determinante de \ud434\u2217 desenvolvido segundo a sua \ud456-e´sima linha e´ exatamente a equac¸a\u2dco (2.10).
A demonstrac¸a\u2dco de (2.11) e´ feita de forma ana´loga, mas usando o item (d) do Teorema 2.13, ou seja,
que det(\ud434) = det(\ud434\ud461). \u25a0
Marc¸o 2010 Reginaldo J. Santos
134 Inversa\u2dco de Matrizes e Determinantes
Teorema 2.18. Se \ud434 e´ uma matriz \ud45b× \ud45b, enta\u2dco
\ud434(adj(\ud434)) = (adj(\ud434))\ud434 = det(\ud434)\ud43c\ud45b
Demonstrac¸a\u2dco. O produto da matriz \ud434 pela matriz adjunta de \ud434 e´ dada por\u23a1
\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a3
\ud44e11 \ud44e12 . . . \ud44e1\ud45b
.
.
. . . .
.
.
.
\ud44e\ud4561 \ud44e\ud4562 . . . \ud44e\ud456\ud45b
.
.
. . . .
.
.
.
\ud44e\ud45b1 \ud44e\ud45b2 . . . \ud44e\ud45b\ud45d
\u23a4
\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a6
\u23a1
\u23a2\u23a2\u23a2\u23a3
\ud44e\u2dc11
\ud44e\u2dc12
.
.
.
\ud44e\u2dc1\ud45b
. . .
. . .
. . .
. . .
\ud44e\u2dc\ud4571
\ud44e\u2dc\ud4572
.
.
.
\ud44e\u2dc\ud457\ud45d
. . .
. . .
. . .
. . .
\ud44e\u2dc\ud45b1
\ud44e\u2dc\ud45b2
.
.
.
\ud44e\u2dc\ud45b\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a6
O elemento de posic¸a\u2dco \ud456, \ud457 de \ud434 adj(\ud434) e´
(\ud434 adj(\ud434))\ud456\ud457 =
\ud45b\u2211
\ud458=1
\ud44e\ud456\ud458\ud44e\u2dc\ud457\ud458 = \ud44e\ud4561\ud44e\u2dc\ud4571 + \ud44e\ud4562\ud44e\u2dc\ud4572 + . . . \ud44e\ud456\ud45b\ud44e\u2dc\ud457\ud45b .
Pelo Lema 2.17, equac¸a\u2dco (2.10) e do Teorema 2.11 na pa´gina 117 segue-se que
(\ud434 adj(\ud434))\ud456\ud457 =
{
det(\ud434) se \ud456 = \ud457
0 se \ud456 \u2215= \ud457.
Matrizes Vetores e Geometria Anal\u131´tica Marc¸o 2010
2.2 Determinantes 135
Assim,
\ud434 adj(\ud434) =
\u23a1
\u23a2\u23a2\u23a2\u23a3
det(\ud434) 0 . . . 0
0 det(\ud434) . . . 0
.
.
. . . .
.
.
.
0 0 . . . det(\ud434)
\u23a4
\u23a5\u23a5\u23a5\u23a6 = det(\ud434)\ud43c\ud45b .
Analogamente, usando Lema 2.17, equac¸a\u2dco (2.11), se prova que adj(\ud434) \ud434 = det(\ud434)\ud43c\ud45b. \u25a0
Exemplo 2.20. Vamos mostrar que se uma matriz \ud434 e´ singular, enta\u2dco adj(\ud434) tambe´m e´ singular.
Vamos separar em dois casos.
(a) Se \ud434 = 0¯, enta\u2dco adj(\ud434) tambe´m e´ a matriz nula, que e´ singular.
(b) Se \ud434 \u2215= 0¯, enta\u2dco pelo Teorema 2.18 na pa´gina 134, adj(\ud434)\ud434 = 0¯. Mas, enta\u2dco, se adj(\ud434) fosse
invert\u131´vel, enta\u2dco \ud434 seria igual a` matriz nula (por que?), que estamos assumindo na\u2dco ser este o
caso. Portanto, adj(\ud434) tem que ser singular.
Corola´rio 2.19. Seja \ud434 uma matriz \ud45b× \ud45b. Se det(\ud434) \u2215= 0, enta\u2dco
\ud434\u22121 =
1
det(\ud434)
adj(\ud434) ;
Marc¸o 2010 Reginaldo J. Santos
136 Inversa\u2dco de Matrizes e Determinantes
Demonstrac¸a\u2dco. Se det(\ud434) \u2215= 0, enta\u2dco definindo \ud435 = 1
det(\ud434)
adj(\ud434), pelo Teorema 2.18 temos que
\ud434\ud435 = \ud434(
1
det(\ud434)
adj(\ud434)) =
1
det(\ud434)
(\ud434 adj(\ud434)) =
1
det(\ud434)
det(\ud434)\ud43c\ud45b = \ud43c\ud45b .
Aqui, usamos a propriedade (j) do Teorema 1.1 na pa´gina 10. Portanto, \ud434 e´ invert\u131´vel e \ud435 e´ a inversa
de \ud434. \u25a0
Exemplo 2.21. No Exemplo 2.17 na pa´gina 127 mostramos como obter rapidamente a inversa de ma
matriz 2× 2. Usando o Corola´rio 2.19 podemos tambe´m obter a inversa de uma matriz 2× 2,
\ud434 =
[
\ud44e \ud44f
\ud450 \ud451
]
,
\ud434\u22121 =
1
det(\ud434)
adj(\ud434) =
1
det(\ud434)
[
\ud451 \u2212\ud44f
\u2212\ud450 \ud44e
]
, se det(\ud434) \u2215= 0
Ou seja, a inversa de uma matriz 2 × 2 e´ facilmente obtida trocando-se a posic¸a\u2dco dos elementos da
diagonal principal, trocando-se o sinal dos outros elementos e dividindo-se todos os elementos pelo
determinante de \ud434.
Exemplo 2.22. Vamos calcular a inversa da matriz
\ud435 =
\u23a1
\u23a3 1 2 30 3 2
0 0 \u22122
\u23a4
\u23a6 .
Matrizes Vetores e Geometria Anal\u131´tica Marc¸o 2010
2.2 Determinantes 137
A sua adjunta foi calculada no Exemplo 2.19 na pa´gina 132. Assim,
\ud435\u22121 =
1
det(\ud435)
adj(\ud435) =
1
\u22126
\u23a1
\u23a3 \u22126 4 \u221250 \u22122 \u22122
0 0 3
\u23a4
\u23a6 =
\u23a1
\u23a3 1 \u221223 560 1
3
1
3
0 0 \u22121
2
\u23a4
\u23a6 .
Corola´rio 2.20 (Regra de Cramer). Se o sistema linear \ud434\ud44b = \ud435 e´ tal que a matriz \ud434 e´ \ud45b × \ud45b e
invert\u131´vel, enta\u2dco a soluc¸a\u2dco do sistema e´ dada por
\ud4651 =
det(\ud4341)
det(\ud434)
, \ud4652 =
det(\ud4342)
det(\ud434)
, . . . , \ud465\ud45b =
det(\ud434\ud45b)
det(\ud434)
,
em que \ud434\ud457 e´ a matriz que se obtem de \ud434 substituindo-se a sua \ud457-e´sima coluna por \ud435, para \ud457 =
1, . . . , \ud45b.
Marc¸o 2010 Reginaldo J. Santos
138 Inversa\u2dco de Matrizes e Determinantes
Demonstrac¸a\u2dco. Como \ud434 e´ invert\u131´vel, pelo Corola´rio 2.19
\ud44b = \ud434\u22121\ud435 =
1
det(\ud434)
adj(\ud434)\ud435.
A entrada \ud465\ud457 e´ dada por
\ud465\ud457 =
1
det(\ud434)
(\ud44e\u2dc1\ud457\ud44f1 + . . .+ \ud44e\u2dc\ud45b\ud457\ud44f\ud45b) =
det(\ud434\ud457)
det(\ud434)
,
em que \ud434\ud457 e´ a matriz que se obtem de \ud434 substituindo-se a sua \ud457-e´sima coluna por \ud435, para \ud457 =
1, . . . , \ud45b e det(\ud434\ud457) foi calculado fazendo o desenvolvimento em cofatores em relac¸a\u2dco a \ud457-e´sima coluna
de \ud434\ud457 . \u25a0
Se a matriz \ud434 na\u2dco e´ invert\u131´vel, enta\u2dco a regra de Cramer na\u2dco pode ser aplicada. Pode ocorrer que
det(\ud434) = det(\ud434\ud457) = 0, para \ud457 = 1, . . . , \ud45b e o sistema na\u2dco tenha soluc¸a\u2dco (verifique!). A regra de
Cramer tem um valor teo´rico, por fornecer uma fo´rmula para a soluc¸a\u2dco de um sistema linear, quando
a matriz do sistema e´ quadrada e invert\u131´vel.
Matrizes Vetores e Geometria Anal\u131´tica Marc¸o 2010
2.2 Determinantes 139
Exerc\u131´cios Nume´ricos (respostas na pa´gina 576)
2.2.1. Se det(\ud434) = \u22123, encontre
(a) det(\ud4342); (b) det(\ud4343); (c) det(\ud434\u22121); (d) det(\ud434\ud461);
2.2.2. Se \ud434 e \ud435 sa\u2dco matrizes \ud45b× \ud45b tais que det(\ud434) = \u22122 e det(\ud435) = 3, calcule det(\ud434\ud461\ud435\u22121).
2.2.3. Seja \ud434 = (\ud44e\ud456\ud457)3×3 tal que det(\ud434) = 3. Calcule o determinante das matrizes a seguir:
(a)
\u23a1
\u23a3 \ud44e11 \ud44e12 \ud44e13 + \ud44e12\ud44e21 \ud44e22 \ud44e23