Prove a parte (c) do Teorema 1.4.2.

Passo 1

O item (c) diz que A - A = A + ( - A ) , vamos encontrar os dois lados da equação e comparar

A = [[[a_11,a_12,cdots,a_1 j],[a_21,a_22,cdots,a_2 j],[vdots,vdots,ddots,,vdots],[a_i 1,a_i 2,cdots,a_i j]]] => A - A = [[[a_11,a_12,cdots,a_1 j],[a_21,a_22,cdots,a_2 j],[vdots,vdots,ddots,,vdots],[a_i 1,a_i 2,cdots,a_i j]]] - [[[a_11,a_12,cdots,a_1 j],[a_21,a_22,cdots,a_2 j],[vdots,vdots,ddots,,vdots],[a_i 1,a_i 2,cdots,a_i j]]] =>

=> A - A = [[[a_11,-,a_11,a_12,-,a_12,cdots,a_1 j,-,a_1 j],[a_21,-,a_21,a_22,-,a_22,cdots,a_2 j,-,a_2 j],[vdots,vdots,ddots,,vdots],[a_i 1,-,a_21,a_i 2,-,a_i 2,cdots,a_i j,-,a_i j]]] = [[[0,0,cdots,0],[0,0,cdots,0],[vdots,vdots,ddots,vdots],[0,0,cdots,0]]]

Agora o outro lado

- A = [[[-,a_11,-,a_12,cdots,-,a_1 j],[-,a_21,-,a_22,cdots,- a_2 j],[vdots,vdots,ddots,,vdots],[- a_i 1,-,a_i 2,cdots,-,a_i j]]] =>

=> A + (- A) = [[[a_11,a_12,cdots,a_1 j],[a_21,a_22,cdots,a_2 j],[vdots,vdots,ddots,,vdots],[a_i 1,a_i 2,cdots,a_i j]]] + [[[-,a_11,-,a_12,cdots,-,a_1 j],[-,a_21,-,a_22,cdots,- a_2 j],[vdots,vdots,ddots,,vdots],[- a_i 1,-,a_i 2,cdots,-,a_i j]]] =>

=> A + (- A) = [[[a_11,+,(,-,a_11,),a_12,+,(,-,a_12,),cdots,a_1 j,+,(,-,a_1 j,)],[a_21,+,(,-,a_21,),a_22,+,(,-,a_22,),cdots,a_2 j,+,(,-,a_2 j,)],[vdots,vdots,ddots,,vdots],[a_i 1,+,(,-,a_21,),a_i 2,+,(,-,a_i 2,),cdots,a_i j,+,(,-,a_i j,)]]] = [[[0,0,cdots,0],[0,0,cdots,0],[vdots,vdots,ddots,vdots],[0,0,cdots,0]]]

Resposta

A + (- A) = [[[a_11,+,(,-,a_11,),a_12,+,(,-,a_12,),cdots,a_1 j,+,(,-,a_1 j,)],[a_21,+,(,-,a_21,),a_22,+,(,-,a_22,),cdots,a_2 j,+,(,-,a_2 j,)],[vdots,vdots,ddots,,vdots],[a_i 1,+,(,-,a_21,),a_i 2,+,(,-,a_i 2,),cdots,a_i j,+,(,-,a_i j,)]]] = [[[0,0,cdots,0],[0,0,cdots,0],[vdots,vdots,ddots,vdots],[0,0,cdots,0]]]

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