Matrizes  Vetores e Geometria Analítica
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Matrizes Vetores e Geometria Analítica


DisciplinaGeometria Analítica e Sistemas Lineares137 materiais832 seguidores
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do Determinante
Vamos provar uma propriedade importante do determinante. Para isso vamos escrever a matriz
\ud434 = (\ud44e\ud456\ud457)\ud45b×\ud45b em termos das suas linhas
\ud434 =
\u23a1
\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a3
\ud4341
.
.
.
\ud434\ud458\u22121
\ud434\ud458
\ud434\ud458+1
.
.
.
\ud434\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a6
,
em que \ud434\ud456 e´ a linha \ud456 da matriz \ud434, ou seja, \ud434\ud456 = [ \ud44e\ud4561 \ud44e\ud4562 . . . \ud44e\ud456\ud45b ]. Se a linha \ud434\ud458 e´ escrita na forma
\ud434\ud458 = \ud6fc\ud44b + \ud6fd\ud44c , em que \ud44b = [ \ud4651 . . . \ud465\ud45b ], \ud44c = [ \ud4661 . . . \ud466\ud45b ] e \ud6fc e \ud6fd sa\u2dco escalares, dizemos que
a linha \ud434\ud458 e´ combinac¸a\u2dco linear de \ud44b e \ud44c . Se a linha \ud434\ud458 e´ combinac¸a\u2dco linear de \ud44b e \ud44c , enta\u2dco o
determinante pode ser decomposto como no resultado seguinte.
Matrizes Vetores e Geometria Anal\u131´tica Marc¸o 2010
2.2 Determinantes 115
Teorema 2.10. Seja \ud434 = (\ud44e\ud456\ud457)\ud45b×\ud45b escrita em termos das suas linhas, denotadas por \ud434\ud456, ou seja,
\ud434\ud456 = [ \ud44e\ud4561 \ud44e\ud4562 . . . \ud44e\ud456\ud45b ]. Se para algum \ud458, a linha \ud434\ud458 = \ud6fc\ud44b + \ud6fd\ud44c , em que \ud44b = [ \ud4651 . . . \ud465\ud45b ],
\ud44c = [ \ud4661 . . . \ud466\ud45b ] e \ud6fc e \ud6fd sa\u2dco escalares, enta\u2dco:
det
\u23a1
\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a3
\ud4341
.
.
.
\ud434\ud458\u22121
\ud6fc\ud44b + \ud6fd\ud44c
\ud434\ud458+1
.
.
.
\ud434\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a6
= \ud6fc det
\u23a1
\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a3
\ud4341
.
.
.
\ud434\ud458\u22121
\ud44b
\ud434\ud458+1
.
.
.
\ud434\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a6
+ \ud6fd det
\u23a1
\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a3
\ud4341
.
.
.
\ud434\ud458\u22121
\ud44c
\ud434\ud458+1
.
.
.
\ud434\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a6
.
Aqui, \ud434\ud458 = \ud6fc\ud44b + \ud6fd\ud44c = [\ud6fc\ud4651 + \ud6fd\ud4661 . . . \ud6fc\ud465\ud45b + \ud6fd\ud466\ud45b ].
Demonstrac¸a\u2dco. Vamos provar aqui somente para \ud458 = 1. Para \ud458 > 1 e´ demonstrado no Ape\u2c6ndice III
na pa´gina 144. Se \ud4341 = \ud6fc\ud44b + \ud6fd\ud44c , em que \ud44b = [ \ud4651 . . . \ud465\ud45b ], \ud44c = [ \ud4661 . . . \ud466\ud45b ] e \ud6fc e \ud6fd sa\u2dco escalares,
Marc¸o 2010 Reginaldo J. Santos
116 Inversa\u2dco de Matrizes e Determinantes
enta\u2dco:
det
\u23a1
\u23a2\u23a2\u23a2\u23a3
\ud6fc\ud44b + \ud6fd\ud44c
\ud4342
.
.
.
\ud434\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a6 =
\ud45b\u2211
\ud457=1
(\u22121)1+\ud457(\ud6fc\ud465\ud457 + \ud6fd\ud466\ud457) det(\ud434\u2dc1\ud457)
= \ud6fc
\ud45b\u2211
\ud457=1
\ud465\ud457 det(\ud434\u2dc1\ud457) + \ud6fd
\ud45b\u2211
\ud457=1
\ud466\ud457 det(\ud434\u2dc1\ud457)
= \ud6fc det
\u23a1
\u23a2\u23a2\u23a2\u23a3
\ud44b
\ud4342
.
.
.
\ud434\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a6+ \ud6fd det
\u23a1
\u23a2\u23a2\u23a2\u23a3
\ud44c
\ud4342
.
.
.
\ud434\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a6
\u25a0
Exemplo 2.12. O ca´lculo do determinante da matriz a seguir pode ser feito da seguinte forma:
det
[
cos \ud461 sen \ud461
2 cos \ud461\u2212 3 sen \ud461 2 sen \ud461+ 3 cos \ud461
]
= 2det
[
cos \ud461 sen \ud461
cos \ud461 sen \ud461
]
+ 3det
[
cos \ud461 sen \ud461
\u2212 sen \ud461 cos \ud461
]
= 3
Pela definic¸a\u2dco de determinante, o determinante deve ser calculado fazendo-se o desenvolvimento
em cofatores segundo a 1a. linha. O pro´ximo resultado, que na\u2dco vamos provar neste momento
Matrizes Vetores e Geometria Anal\u131´tica Marc¸o 2010
2.2 Determinantes 117
(Ape\u2c6ndice III na pa´gina 144), afirma que o determinante pode ser calculado fazendo-se o desen-
volvimento em cofatores segundo qualquer linha ou qualquer coluna.
Teorema 2.11. Seja \ud434 uma matriz \ud45b × \ud45b. O determinante de \ud434 pode ser calculado fazendo-se o
desenvolvimento em cofatores segundo qualquer linha ou qualquer coluna.
det(\ud434) = \ud44e\ud4561\ud44e\u2dc\ud4561 + \ud44e\ud4562\ud44e\u2dc\ud4562 + . . .+ \ud44e\ud456\ud45b\ud44e\u2dc\ud456\ud45b =
\ud45b\u2211
\ud457=1
\ud44e\ud456\ud457 \ud44e\u2dc\ud456\ud457, para \ud456 = 1, . . . , \ud45b, (2.8)
= \ud44e1\ud457 \ud44e\u2dc1\ud457 + \ud44e2\ud457 \ud44e\u2dc2\ud457 + . . .+ \ud44e\ud45b\ud457 \ud44e\u2dc\ud45b\ud457 =
\ud45b\u2211
\ud456=1
\ud44e\ud456\ud457 \ud44e\u2dc\ud456\ud457, para \ud457 = 1, . . . , \ud45b, (2.9)
em que \ud44e\u2dc\ud456\ud457 = (\u22121)\ud456+\ud457 det(\ud434\u2dc\ud456\ud457) e´ o cofator do elemento \ud44e\ud456\ud457 . A expressa\u2dco (2.8) e´ chamada desen-
volvimento em cofatores do determinante de \ud434 em termos da \ud456-e´sima linha e (2.9) e´ chamada
desenvolvimento em cofatores do determinante de \ud434 em termos da \ud457-e´sima coluna.
Marc¸o 2010 Reginaldo J. Santos
118 Inversa\u2dco de Matrizes e Determinantes
Temos a seguinte consequ¨e\u2c6ncia deste resultado.
Corola´rio 2.12. Seja \ud434 uma matriz \ud45b× \ud45b. Se \ud434 possui duas linhas iguais, enta\u2dco det(\ud434) = 0.
Demonstrac¸a\u2dco. O resultado e´ claramente verdadeiro para matrizes 2× 2. Supondo que o resultado
seja verdadeiro para matrizes (\ud45b \u2212 1) × (\ud45b \u2212 1), vamos provar que ele e´ verdadeiro para matrizes
\ud45b× \ud45b. Suponhamos que as linhas \ud458 e \ud459 sejam iguais, para \ud458 \u2215= \ud459. Desenvolvendo o determinante de
\ud434 em termos de uma linha \ud456, com \ud456 \u2215= \ud458, \ud459, obtemos
det(\ud434) =
\ud45b\u2211
\ud457=1
\ud44e\ud456\ud457 \ud44e\u2dc\ud456\ud457 =
\ud45b\u2211
\ud457=1
(\u22121)\ud456+\ud457\ud44e\ud456\ud457 det(\ud434\u2dc\ud456\ud457).
Mas, cada \ud434\u2dc\ud456\ud457 e´ uma matriz (\ud45b\u2212 1)× (\ud45b\u2212 1) com duas linhas iguais. Como estamos supondo que o
resultado seja verdadeiro para estas matrizes, enta\u2dco det(\ud434\u2dc\ud456\ud457) = 0. Isto implica que det(\ud434) = 0. \u25a0
No pro´ximo resultado mostramos como varia o determinante de uma matriz quando aplicamos
operac¸o\u2dces elementares sobre suas linhas.
Teorema 2.13. Sejam \ud434 e \ud435 matrizes \ud45b× \ud45b.
(a) Se \ud435 e´ obtida de \ud434 multiplicando-se uma linha por um escalar \ud6fc, enta\u2dco
det(\ud435) = \ud6fc det(\ud434) ;
Matrizes Vetores e Geometria Anal\u131´tica Marc¸o 2010
2.2 Determinantes 119
(b) Se \ud435 resulta de \ud434 pela troca da posic¸a\u2dco de duas linhas \ud458 \u2215= \ud459, enta\u2dco
det(\ud435) = \u2212 det(\ud434) ;
(c) Se \ud435 e´ obtida de \ud434 substituindo a linha \ud459 por ela somada a um mu´ltiplo escalar de uma linha \ud458,
\ud458 \u2215= \ud459, enta\u2dco
det(\ud435) = det(\ud434) .
Demonstrac¸a\u2dco. (a) Segue diretamente do Teorema 2.10 na pa´gina 115.
(b) Sejam
\ud434 =
\u23a1
\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a3
\ud4341
.
.
.
\ud434\ud458
.
.
.
\ud434\ud459
.
.
.
\ud434\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a6
e \ud435 =
\u23a1
\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a3
\ud4341
.
.
.
\ud434\ud459
.
.
.
\ud434\ud458
.
.
.
\ud434\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a6
.
Marc¸o 2010 Reginaldo J. Santos
120 Inversa\u2dco de Matrizes e Determinantes
Agora, pelo Teorema 2.10 na pa´gina 115 e o Corola´rio 2.12, temos que
0 = det
\u23a1
\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a3
\ud4341
.
.
.
\ud434\ud458 + \ud434\ud459
.
.
.
\ud434\ud458 + \ud434\ud459
.
.
.
\ud434\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a6
= det
\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
+ det
\u23a1
\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a3
\ud4341
.
.
.
\ud434\ud458
.
.
.
\ud434\ud459
.
.
.
\ud434\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a6
+ det
\u23a1
\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a3
\ud4341
.
.
.
\ud434\ud459
.
.
.
\ud434\ud458
.
.
.
\ud434\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a6
+ det
\u23a1
\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a3
\ud4341
.
.
.
\ud434\ud459
.
.
.
\ud434\ud459
.
.
.
\ud434\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a6
= 0 + det(\ud434) + det(\ud435) + 0.
Portanto, det(\ud434) = \u2212 det(\ud435).
(c) Novamente, pelo Teorema 2.10 na pa´gina 115, temos que
det
\u23a1
\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a3
\ud4341
.
.
.
\ud434\ud458
.
.
.
\ud434\ud459 + \ud6fc\ud434\ud458
.
.
.
\ud434\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a6
= det
\u23a1
\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a3
\ud4341
.
.
.
\ud434\ud458
.
.
.
\ud434\ud459
.
.
.
\ud434\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a6
+ \ud6fc det
\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
= det
\u23a1
\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a2\u23a3
\ud4341
.
.
.
\ud434\ud458
.
.
.
\ud434\ud459
.
.
.
\ud434\ud45b
\u23a4
\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a5\u23a6
.
\u25a0
Matrizes Vetores e Geometria Anal\u131´tica Marc¸o 2010
2.2 Determinantes 121
Exemplo 2.13. Vamos calcular o determinante da matriz
\ud434 =
\u23a1
\u23a3 0 1 53 \u22126 9
2 6 1
\u23a4
\u23a6
usando operac¸o\u2dces elementares para transforma´-la numa matriz triangular superior e aplicando o Te-
orema 2.13.
det(\ud434) = \u2212 det
\u23a1
\u23a3 3 \u22126 90 1 5
2 6 1
\u23a4
\u23a6 1a. linha \u2190\u2192 2a. linha
= \u22123 det
\u23a1
\u23a3 1 \u22122 30 1 5
2 6 1
\u23a4
\u23a6 1/3×1a. linha \u2212\u2192 1a. linha
= \u22123 det
\u23a1
\u23a3 1 \u22122 30 1 5
0 10 \u22125
\u23a4
\u23a6 \u22122×1a. linha+3a. linha \u2212\u2192 3a. linha
= \u22123 det
\u23a1
\u23a3 1 \u22122 30 1 5
0 0 \u221255
\u23a4
\u23a6 \u221210×2a. linha+3a. linha \u2212\u2192 3a. linha
= (\u22123)(\u221255) = 165
Quando multiplicamos uma linha de uma matriz por um escalar \ud6fc o determinante da nova matriz e´
igual a \ud6fc multiplicado pelo determinante da matriz antiga. Mas o que estamos calculando aqui e´ o
determinante da matriz antiga, por isso ele e´ igual a 1/\ud6fc multiplicado pelo determinante da matriz
nova.
Marc¸o 2010 Reginaldo J. Santos
122 Inversa\u2dco de Matrizes e Determinantes
Para se calcular o determinante de uma matriz \ud45b × \ud45b pela expansa\u2dco em cofatores, precisamos
fazer \ud45b produtos e calcular \ud45b determinantes de matrizes (\ud45b \u2212 1) × (\ud45b \u2212 1), que por sua vez vai
precisar de \ud45b \u2212 1 produtos e assim por diante. Portanto, ao todo sa\u2dco necessa´rios da ordem de \ud45b!
produtos. Para se calcular o determinante de uma matriz 20× 20, e´ necessa´rio se realizar 20! \u2248 1018
produtos. Os computadores pessoais realizam da ordem de 108 produtos por segundo. Portanto, um
computador pessoal precisaria de cerca de 1010 segundos ou 103 anos para calcular o determinante
de uma matriz 20×20 usando a expansa\u2dco em cofatores. Entretanto usando o me´todo apresentado no
exemplo anterior para o ca´lculo do determinante,