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
.