Verifique que A^3 = overset(-)(0) , para

A = [[[0,1,0],[0,0,1],[0,0,0]]] .

Passo 1

Sabemos que A^3 é o mesmo que A xx A xx A , dessa forma, podemos aplicar as propriedades de operações com matrizes e afirmar que:

A^3 = ( A xx A ) xx A

Realizando a multiplicação (A xx A) :

A xx A = [[[0,*,0,+,1,*,0,+,0,*,0,0,*,1,+,1,*,0,+,0,*,0,0,*,0,+,1,*,1,+,0,*,0],[0,*,0,+,0,*,0,+,1,*,0,0,*,1,+,0,*,0,+,1,*,0,0,*,0,+,0,*,1,+,1,*,0],[0,*,0,+,0,*,0,+,0,*,0,0,*,1,+,0,*,0,+,0,*,0,0,*,0,+,0,*,1,+,0,*,0]]]

A xx A = [[[0,+,0,+,0,0,+,0,+,0,0,+,1,+,0],[0,+,0,+,0,0,+,0,+,0,0,+,0,+,0],[0,+,0,+,0,0,+,0,+,0,0,+,0,+,0]]]

A xx A = [[[0,0,1],[0,0,0],[0,0,0]]]

Passo 2

Em seguida, realizando a multiplicação de ( A xx A ) por A:

(A xx A) xx A = [[[0,*,0,+,0,*,0,+,1,*,0,0,*,1,+,0,*,0,+,1,*,0,0,*,0,+,0,*,1,+,1,*,0],[0,*,0,+,0,*,0,+,0,*,0,0,*,1,+,0,*,0,+,0,*,0,0,*,0,+,0,*,1,+,0,*,0],[0,*,0,+,0,*,0,+,0,*,0,0,*,1,+,0,*,0,+,0,*,0,0,*,0,+,0,*,1,+,0,*,0]]]

(A xx A) xx A = [[[0,+,0,+,0,0,+,0,+,0,0,+,0,+,0],[0,+,0,+,0,0,+,0,+,0,0,+,0,+,0],[0,+,0,+,0,0,+,0,+,0,0,+,0,+,0]]]

(A xx A) xx A = [[[0,0,0],[0,0,0],[0,0,0]]]

Passo 3

Como foi visto que A^3 = ( A xx A ) xx A , e

(A xx A) xx A = [[[0,0,0],[0,0,0],[0,0,0]]]

Que pode ser reescrito como (A xx A) xx A = overset(-)(0) .

Então A^3 = overset(-)(0) .

Resposta

(A xx A) xx A = [[[0,0,0],[0,0,0],[0,0,0]]]

Então A^3 = overset(-)(0)

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