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Enunciado

Álgebra Linear e Geometria Analítica - Anton - Álgebra Linear- Ed: 10º - Capítulo 1.6 - Ex. 6

[[-,x,-,2,y,-,3,z,=,0],[w,+,x,+,4,y,+,4,z,=,7],[w,+,3,x,+,7,y,+,9,z,=,4],[-,w,-,2,x,-,4,y,-,6,z,=,6]]

Passo 1

O enunciado fala pra gente resolver esse sistema invertendo a matriz de coeficientes e usando o Teorema 1.6.2 que diz que “Se A dor uma matriz invertível n xx n , então para cada matriz b de tamanho n xx 1 , o sistema de equações A x = b tem exatamente uma solução, a saber, x = A^(- 1) b ”. Então bora aplicar isso ai no nossos sistema.

Passo 2

Temos que

[[-,x,-,2,y,-,3,z,=,0],[w,+,x,+,4,y,+,4,z,=,7],[w,+,3,x,+,7,y,+,9,z,=,4],[-,w,-,2,x,-,4,y,-,6,z,=,6]] => [[[0,-,1,-,2,-,3],[1,1,4,4],[1,3,7,9],[-,1,-,2,-,4,-,6]]] [[[w],[x],[y],[z]]] = [[[0],[7],[4],[6]]] =>

=> underset(I)(underset(⏝)([[[0,-,1,-,2,-,3],[1,1,4,4],[1,3,7,9],[-,1,-,2,-,4,-,6]]]^(- 1) [[[0,-,1,-,2,-,3],[1,1,4,4],[1,3,7,9],[-,1,-,2,-,4,-,6]]])) [[[w],[x],[y],[z]]] = [[[0,-,1,-,2,-,3],[1,1,4,4],[1,3,7,9],[-,1,-,2,-,4,-,6]]]^(- 1) [[[0],[7],[4],[6]]] =>

=> [[[w],[x],[y],[z]]] = [[[0,-,1,-,2,-,3],[1,1,4,4],[1,3,7,9],[-,1,-,2,-,4,-,6]]]^(- 1) [[[0],[7],[4],[6]]]

Agora nossa matriz é de ordem maior que 2, bora calcular a inversa dela através do algoritmo.

Passo 3

Nosso algoritmo diz pra operarmos [ A | I ] até chegarmos em [ I | X ] , onde X = A^(- 1) . Então bora lá

A = [[[0,-,1,-,2,-,3],[1,1,4,4],[1,3,7,9],[-,1,-,2,-,4,-,6]]] => [A,I] = [[[[[0,-,1,-,2,-,3],[1,1,4,4],[1,3,7,9],[-,1,-,2,-,4,-,6]]]],[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]] =>

=> [[L_1,<=>,L_2],[],[],[]] = [[[[[1,1,4,4],[0,-,1,-,2,-,3],[1,3,7,9],[-,1,-,2,-,4,-,6]]]],[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]] =>

=> [[],[,L_2,=,-,L_2],[,L_3,=,L_3,-,L_1],[,L_4,=,L_4,+,L_1]] = [[[[[1,1,4,4],[0,1,2,3],[0,2,3,5],[0,-,1,0,-,2]]]],[[0,1,0,0],[-,1,0,0,0],[0,-,1,1,0],[0,1,0,1]]] =>

=> [[,L_1,=,L_1,+,L_4],[,],[,L_3,=,L_3,+,2 L_4],[,]] = [[[[[1,0,4,2],[0,1,2,3],[0,0,3,1],[0,-,1,0,-,2]]]],[[0,2,0,1],[-,1,0,0,0],[0,1,1,2],[0,1,0,1]]] =>

=> [[,],[,],[,],[,,L_4,=,L_4,+,L_2]] = [[[[[1,0,4,2],[0,1,2,3],[0,0,3,1],[0,0,2,1]]]],[[0,2,0,1],[-,1,0,0,0],[0,1,1,2],[-,1,1,0,1]]] =>

=> [[,],[,],[,,L_3,=,L_3,-,L_4],[,,]] = [[[[[1,0,4,2],[0,1,2,3],[0,0,1,0],[0,0,2,1]]]],[[0,2,0,1],[-,1,0,0,0],[1,0,1,1],[-,1,1,0,1]]] =>

=> [[,],[,],[,,],[,,,L_4,=,L_4,-,2,L_3]] = [[[[[1,0,4,2],[0,1,2,3],[0,0,1,0],[0,0,0,1]]]],[[0,2,0,1],[-,1,0,0,0],[1,0,1,1],[-,3,1,-,2,-,1]]] =>

=> [[,,L_1,=,L_1,-,4,L_3],[,,L_2,=,L_2,-,2,L_3],[,,],[,,,]] = [[[[[1,0,0,2],[0,1,0,3],[0,0,1,0],[0,0,0,1]]]],[[-,4,2,-,4,-,3],[-,3,0,-,2,-,2],[1,0,1,1],[-,3,1,-,2,-,1]]] =>

=> [[,,L_1,=,L_1,-,2,L_4],[,,L_2,=,L_2,-,3,L_4],[,,],[,,,]] = [[[[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]]],[[2,0,0,-,1],[6,-,3,4,1],[1,0,1,1],[-,3,1,-,2,-,1]]] =>

Logo

A^(- 1) = [[[[[[[2,0,0,-,1],[6,-,3,4,1],[1,0,1,1],[-,3,1,-,2,-,1]]]]]]]

Passo 4

Agora que encontramos a inversa da nossa matriz podemos finalmente resolver a equação

[[[w],[x],[y],[z]]] = [[[0,-,1,-,2,-,3],[1,1,4,4],[1,3,7,9],[-,1,-,2,-,4,-,6]]]^(- 1) [[[0],[7],[4],[6]]] =>

=> [[[w],[x],[y],[z]]] = [[[2,0,0,-,1],[6,-,3,4,1],[1,0,1,1],[-,3,1,-,2,-,1]]] [[[0],[7],[4],[6]]] = [[w,=,0,+,0,+,0,-,6,=,-,6],[,,x,=,0,-,21,+,16,+,6,=,1],[,,y,=,0,+,0,+,4,+,6,=,10],[,,z,=,0,+,7,-,8,-,6,=,-,7]]

Logo w = - 6 , x = 1 , y = 10 e z = - 7

Resposta

w = - 6 , x = 1 , y = 10 e z = - 7

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