Enunciado

Dadas A = 2 - 3 - 5 - 1 4 5 1 - 3 - 4 , B = - 1 3 5 1 - 3 - 5 - 1 3 5 e C = 2 - 2 - 4 - 1 3 4 1 - 2 - 3 ,

  1. Mostre que A B = B A = 0 , A C = A e C A = C .
  2. Use os resultados de (a) para mostra que A C B = C B A , A 2 - B 2 = A - B A + B e A ± B 2 = A 2 + B 2 .

Passo 1

  1. Para mostre que A B = B A = 0 vamos realizar as duas multiplicações de matrizes, para calcular a multiplicação de matrizes pegamos cada linha da primeira matriz com cada coluna da segunda e multiplicamos cada elemento com seu par e somamos, assim:
  2. Primeiro para A B :

    A ⋅ B = 2 - 3 - 5 - 1 4 5 1 - 3 - 4 ⋅ - 1 3 5 1 - 3 - 5 - 1 3 5 =

    = 2 ⋅ - 1 + - 3 ⋅ 1 + - 5 ⋅ - 1 2 ⋅ 3 + - 3 ⋅ - 3 + - 5 ⋅ 3 2 ⋅ 5 + - 3 ⋅ - 5 + - 5 ⋅ 5 - 1 ⋅ - 1 + 4 ⋅ 1 + 5 ⋅ - 1 - 1 ⋅ 3 + 4 ⋅ - 3 + 5 ⋅ 3 - 1 ⋅ 5 + 4 ⋅ - 5 + 5 ⋅ 5 1 ⋅ - 1 + - 3 ⋅ 1 + - 4 ⋅ - 1 1 ⋅ 3 + - 3 ⋅ - 3 + - 4 ⋅ 3 1 ⋅ 5 + - 3 ⋅ - 5 + - 4 ⋅ 5

    = - 2 - 3 + 5 6 + 9 - 15 10 + 15 - 2 5 1 + 4 - 5 - 3 - 12 + 15 - 5 - 20 + 25 - 1 - 3 + 4 3 + 9 - 12 5 + 15 - 20 = 0 0 0 0 0 0 0 0 0 = 0

    Passo 2

    Agora para B A :

    B ⋅ A = - 1 3 5 1 - 3 - 5 - 1 3 5 ⋅ 2 - 3 - 5 - 1 4 5 1 - 3 - 4 =

    = - 1 ⋅ 2 + 3 ⋅ - 1 + 5 ⋅ 1 - 1 ⋅ - 3 + 3 ⋅ 4 + 5 ⋅ - 3 - 1 ⋅ - 5 + 3 ⋅ 5 + 5 ⋅ - 4 1 ⋅ 2 + - 3 ⋅ - 1 + - 5 ⋅ 1 1 ⋅ - 3 + - 3 ⋅ 4 + - 5 ⋅ - 3 1 ⋅ - 5 + - 3 ⋅ 5 + - 5 ⋅ - 4 - 1 ⋅ 2 + 3 ⋅ - 1 + 5 ⋅ 1 - 1 ⋅ - 3 + 3 ⋅ 4 + 5 ⋅ - 3 - 1 ⋅ - 5 + 3 ⋅ 5 + 5 ⋅ - 4

    = - 2 - 3 + 5 3 + 12 - 15 5 + 15 - 20 2 + 3 - 5 - 3 - 12 + 15 - 5 - 15 + 20 - 2 - 3 + 5 3 + 12 - 15 5 + 15 - 20 = 0 0 0 0 0 0 0 0 0 = 0

    Assim, temos que A B = B A .

    Passo 3

    Agora vamos mostrar que A C = A :

    A ⋅ C = 2 - 3 - 5 - 1 4 5 1 - 3 - 4 ⋅ 2 - 2 - 4 - 1 3 4 1 - 2 - 3 =

    = 2 ⋅ 2 + - 3 ⋅ - 1 + - 5 ⋅ 1 2 ⋅ - 2 + - 3 ⋅ 3 + - 5 ⋅ - 2 2 ⋅ - 4 + - 3 ⋅ 4 + - 5 ⋅ - 3 - 1 ⋅ 2 + 4 ⋅ - 1 + 5 ⋅ 1 - 1 ⋅ - 2 + 4 ⋅ 3 + 5 ⋅ - 2 - 1 ⋅ - 4 + 4 ⋅ 4 + 5 ⋅ - 3 1 ⋅ 2 + - 3 ⋅ - 1 + - 4 ⋅ 1 1 ⋅ - 2 + - 3 ⋅ 3 + - 4 ⋅ - 2 1 ⋅ - 4 + - 3 ⋅ 4 + - 4 ⋅ - 3

    = 4 + 3 - 5 - 4 - 9 + 10 - 8 - 12 + 15 - 2 - 4 + 5 2 + 12 - 10 4 + 16 - 15 2 + 3 - 4 - 2 - 9 + 8 - 4 - 12 + 12 = 2 - 3 - 5 - 1 4 5 1 - 3 - 4

    Assim, temos que A C = A .

    Passo 4

    Agora vamos mostrar que C A = C :

    C ⋅ A = 2 - 2 - 4 - 1 3 4 1 - 2 - 3 ⋅ 2 - 3 - 5 - 1 4 5 1 - 3 - 4 =

    = 2 ⋅ 2 + - 2 ⋅ - 1 + - 4 ⋅ 1 2 ⋅ - 3 + - 2 ⋅ 4 + - 4 ⋅ - 3 2 ⋅ - 5 + - 2 ⋅ 5 + - 4 ⋅ - 4 - 1 ⋅ 2 + 3 ⋅ - 1 + 4 ⋅ 1 - 1 ⋅ - 3 + 3 ⋅ 4 + 4 ⋅ - 3 - 1 ⋅ - 5 + 3 ⋅ 5 + 4 ⋅ - 4 1 ⋅ 2 + - 2 ⋅ - 1 + - 3 ⋅ 1 1 ⋅ - 3 + - 2 ⋅ 4 + - 3 ⋅ - 3 1 ⋅ - 5 + - 2 ⋅ 5 + - 3 ⋅ - 4

    = 4 + 2 - 4 - 6 - 8 + 12 - 10 - 10 + 16 - 2 - 3 + 4 3 + 12 - 12 5 + 15 - 16 2 + 2 - 3 - 3 - 8 + 9 - 5 - 10 + 12 = 2 - 2 - 4 - 1 3 4 1 - 2 - 3

    Assim, temos que C A = C .

    Passo 5

  3. Agora vamos mostrar que A C B = C B A :

A C ⋅ B = 2 - 3 - 5 - 1 4 5 1 - 3 - 4 ⋅ - 1 3 5 1 - 3 - 5 - 1 3 5 =

= 2 ⋅ - 1 + - 3 ⋅ 1 + - 5 ⋅ - 1 2 ⋅ 3 + - 3 ⋅ - 3 + - 5 ⋅ 3 2 ⋅ 5 + - 3 ⋅ - 5 + - 5 ⋅ 5 - 1 ⋅ - 1 + 4 ⋅ 1 + 5 ⋅ - 1 - 1 ⋅ 3 + 4 ⋅ - 3 + 5 ⋅ 3 - 1 ⋅ 5 + 4 ⋅ - 5 + 5 ⋅ 5 1 ⋅ - 1 + - 3 ⋅ 1 + - 4 ⋅ - 1 1 ⋅ 3 + - 3 ⋅ - 3 + - 4 ⋅ 3 1 ⋅ 5 + - 3 ⋅ - 5 + - 4 ⋅ 5

= - 2 - 3 + 5 6 + 9 - 15 10 + 15 - 25 1 + 4 - 5 - 3 - 12 + 15 - 5 - 20 + 25 - 1 - 3 + 4 3 + 9 - 12 5 + 15 - 20 = 0 0 0 0 0 0 0 0 0

C ⋅ B A = 2 - 2 - 4 - 1 3 4 1 - 2 - 3 ⋅ 0 =

= 2 ⋅ 0 - 2 ⋅ 0 - 4 ⋅ 0 - 1 ⋅ 0 3 ⋅ 0 4 ⋅ 0 1 ⋅ 0 - 2 ⋅ 0 - 3 ⋅ 0 = 0 0 0 0 0 0 0 0 0

Assim, temos que A C B = C B A .

Passo 2

Agora vamos mostrar que A 2 - B 2 = A - B A + B :

Do lado esquerdo da igualdade temos:

A 2 = 2 - 3 - 5 - 1 4 5 1 - 3 - 4 ⋅ 2 - 3 - 5 - 1 4 5 1 - 3 - 4 =

= 2 ⋅ 2 + - 3 ⋅ - 1 + - 5 ⋅ 1 2 ⋅ - 3 + - 3 ⋅ 4 + - 5 ⋅ - 3 2 ⋅ - 5 + - 3 ⋅ 5 + - 5 ⋅ - 4 - 1 ⋅ 2 + 4 ⋅ - 1 + 5 ⋅ 1 - 1 ⋅ - 3 + 4 ⋅ 4 + 5 ⋅ - 3 - 1 ⋅ - 5 + 4 ⋅ 5 + 5 ⋅ - 4 1 ⋅ 2 + - 3 ⋅ - 1 + - 4 ⋅ 1 1 ⋅ - 3 + - 3 ⋅ 4 + - 4 ⋅ - 3 1 ⋅ - 5 + - 3 ⋅ 5 + - 4 ⋅ - 4

= 4 + 3 - 5 - 6 - 12 + 15 - 10 - 15 + 20 - 2 - 4 + 5 3 + 16 - 15 5 + 20 - 20 2 + 3 - 4 - 3 - 12 + 12 - 5 - 15 + 16 = 2 - 3 - 5 - 1 4 5 1 - 3 - 4

B 2 = - 1 3 5 1 - 3 - 5 - 1 3 5 ⋅ - 1 3 5 1 - 3 - 5 - 1 3 5 =

- 1 ⋅ - 1 + 3 ⋅ 1 + 5 ⋅ - 1 - 1 ⋅ 3 + 3 ⋅ - 3 + 5 ⋅ 3 - 1 ⋅ 5 + 3 ⋅ - 5 + 5 ⋅ 5 1 ⋅ - 1 + - 3 ⋅ 1 + - 5 ⋅ - 1 1 ⋅ 3 + - 3 ⋅ - 3 + - 5 ⋅ 3 1 ⋅ 5 + - 3 ⋅ - 5 + - 5 ⋅ 5 - 1 ⋅ - 1 + 3 ⋅ 1 + 5 ⋅ - 1 - 1 ⋅ 3 + 3 ⋅ - 3 + 5 ⋅ 3 - 1 ⋅ 5 + 3 ⋅ - 5 + 5 ⋅ 5

= 1 + 3 - 5 - 3 - 9 + 15 - 5 - 15 + 25 - 1 - 3 + 5 3 + 9 - 15 5 + 15 - 25 1 + 3 - 5 - 3 - 9 + 15 - 5 - 15 + 25 = - 1 3 5 1 - 3 - 5 - 1 3 5

A 2 - B 2 = 2 - 3 - 5 - 1 4 5 1 - 3 - 4 - - 1 3 5 1 - 3 - 5 - 1 3 5 = 3 - 6 - 10 - 2 7 10 2 - 6 - 9

Do lado direito da igualdade temos:

A - B = 2 - 3 - 5 - 1 4 5 1 - 3 - 4 - - 1 3 5 1 - 3 - 5 - 1 3 5 = 3 - 6 - 10 - 2 7 10 2 - 6 - 9

A + B = 2 - 3 - 5 - 1 4 5 1 - 3 - 4 + - 1 3 5 1 - 3 - 5 - 1 3 5 = 1 0 0 0 1 0 0 0 1

A - B A + B = 3 - 6 - 10 - 2 7 10 2 - 6 - 9 ⋅ 1 0 0 0 1 0 0 0 1 =

= 3 ⋅ 1 + - 6 ⋅ 0 + - 10 ⋅ 0 3 ⋅ 0 + - 6 ⋅ 1 + - 10 ⋅ 0 3 ⋅ 0 + - 6 ⋅ 0 + - 10 ⋅ 1 - 2 ⋅ 1 + 7 ⋅ 0 + 10 ⋅ 0 - 2 ⋅ 0 + 7 ⋅ 1 + 10 ⋅ 0 - 2 ⋅ 0 + 7 ⋅ 0 + 10 ⋅ 1 2 ⋅ 1 + - 6 ⋅ 0 + - 9 ⋅ 0 2 ⋅ 0 + - 6 ⋅ 1 + - 9 ⋅ 0 2 ⋅ 0 + - 6 ⋅ 0 + - 9 ⋅ 1 = 3 - 6 - 10 - 2 7 10 2 - 6 - 9

Assim, temos que A 2 - B 2 = A - B A + B .

Passo 3

Agora vamos mostrar que A ± B 2 = A 2 + B 2 :

Do lado esquerdo da igualdade temos:

A + B 2 = 1 0 0 0 1 0 0 0 1 ⋅ 1 0 0 0 1 0 0 0 1 =

= 1 + 0 + 0 0 + 0 + 0 0 + 0 + 00 0 + 0 + 0 0 + 1 + 0 0 + 0 + 0 0 + 0 + 0 0 + 0 + 0 0 + 0 + 1 = 1 0 0 0 1 0 0 0 1

A - B 2 = 3 - 6 - 10 - 2 7 10 2 - 6 - 9 ⋅ 3 - 6 - 10 - 2 7 10 2 - 6 - 9 =

= 3 ⋅ 3 + - 6 ⋅ - 2 + - 10 ⋅ 2 - 18 - 42 + 60 3 ⋅ - 10 + - 6 ⋅ 10 + - 10 ⋅ - 9 - 2 ⋅ 3 + 7 ⋅ - 2 + 10 ⋅ 2 12 + 49 - 60 - 2 ⋅ - 10 + 7 ⋅ 10 + 10 ⋅ - 9 2 ⋅ 3 + - 6 ⋅ - 2 + - 9 ⋅ 2 - 12 - 42 + 54 2 ⋅ - 10 + - 6 ⋅ 10 + - 9 ⋅ - 9

= 9 + 12 - 20 - 18 - 42 + 60 - 30 - 60 + 90 - 6 - 14 + 20 12 + 49 - 60 20 + 70 - 90 6 + 12 - 18 - 12 - 42 + 54 - 20 - 60 + 81 = 1 0 0 0 1 0 0 0 1

A 2 + B 2 = 2 - 3 - 5 - 1 4 5 1 - 3 - 4 + - 1 3 5 1 - 3 - 5 - 1 3 5 = 1 0 0 0 1 0 0 0 1

Assim, temos que A ± B 2 = A 2 + B 2 .

Resposta

Ei, a resposta está no passo a passo :)

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