[[[1,2,0],[2,1,2],[0,2,1]]]

Passo 1

Show de bolinhas pessoal, o segredo aqui é a gente aplicar o algoritmo de inversibilidade onde operamos [ A | I ] até virar [ I | A^(- 1) ] .

Passo 2

A matriz que temos que inverter é

A = [[[1,2,0],[2,1,2],[0,2,1]]]

[A,I] = [[[1,2,0],[2,1,2],[0,2,1]],[[1,0,0],[0,1,0],[0,0,1]]] => [[],[L_2,=,L_2,-,2,L_1],[]] = [[[1,2,0],[0,-,3,2],[0,2,1]],[[1,0,0],[-,2,1,0],[0,0,1]]] =>

=> [[],[L_2,=,-,(1)/(3) L_2],[L_3,=,(1)/(2) L_3,]] = [[[1,2,0],[0,1,-,(2)/(3)],[0,1,(1)/(2)]],[[1,0,0],[(2)/(3),-,(1)/(3),0],[0,0,(1)/(2)]]] =>

=> [[],[[[],[],[,L_3,=,L_3,-,L_2]],=,[[[1,2,0],[0,1,-,(2)/(3)],[0,0,(7)/(6)]],[[1,0,0],[(2)/(3),-,(1)/(3),0],[-,(2)/(3),(1)/(3),(1)/(2)]]]],[,]] =>

=> [[],[],[L_3,=,(6)/(7) L_3,]] = [[[1,2,0],[0,1,-,(2)/(3)],[0,0,1]],[[1,0,0],[(2)/(3),-,(1)/(3),0],[-,(4)/(7),(2)/(7),(3)/(7)]]] =>

=> [[],[L_2,=,L_2,+,(2)/(3),L_3],[]] = [[[1,2,0],[0,1,0],[0,0,1]],[[1,0,0],[(2)/(7),-,(1)/(7),(2)/(7)],[-,(4)/(7),(2)/(7),(3)/(7)]]] =>

=> [[L_1,=,L_1,-,2,L_2],[],[]] = [[[1,0,0],[0,1,0],[0,0,1]],[[(3)/(7),(2)/(7),-,(4)/(7)],[(2)/(7),-,(1)/(7),(2)/(7)],[-,(4)/(7),(2)/(7),(3)/(7)]]]

Logo

A^(- 1) = [[[(3)/(7),(2)/(7),-,(4)/(7)],[(2)/(7),-,(1)/(7),(2)/(7)],[-,(4)/(7),(2)/(7),(3)/(7)]]]

Resposta

A^(- 1) = [[[(3)/(7),(2)/(7),-,(4)/(7)],[(2)/(7),-,(1)/(7),(2)/(7)],[-,(4)/(7),(2)/(7),(3)/(7)]]]

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