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Question
If A = `[(2, -3),(5, -4),(-6, 1)], "B" = [(-1, 2),(2, 2),(0, 3)] "and C" = [(4, 3),(-1, 4),(-2, 1)]`, Show that (A + B) + C = A + (B + C)
Solution
(A + B) + C = `{[(2, -3),(5, -4),(-6, 1)] + [(-1, 2),(2, 2),(0, 3)]} + [(4, 3),(-1, 4),(-2, 1)]`
= `[(2 - 1, -3 + 2),(5 + 2, -4 + 2),(-6 + 0, 1 + 3)] + [(4, 3),(-1, 4),(-2, 1)]`
= `[(1, -1),(7, -2),(-6, 4)] + [(4, 3),(-1, 4),(-2, 1)]`
= `[(1+ 4, -1 +3),(7 - 1, -2 + 4),(-6 - 2, 4 + 1)]`
∴ (A+ B) + C = `[(5, 2),(6, 2),(-8, 5)]` ....(i)
A + (B + C) = `[(2, -3),(5, -4),(-6, 1)] + {[(-1, 2),(2, 2),(0, )] + [(4, 3),(-1, 4),(-2, 1)]}`
= `[(2, -3),(5, -4),(-6, 1)] + [(-1 + 4, 2 + 3),(2 - 1, 2 + 4),(0 - 2, 3 + 1)]`
= `[(2, -3),(5, -4),(-6, 1)] [(3, 5),(1, 6),(-2, 4)]`
= `[(2 + 3, -3 + 5),(5 + 1, -4 + 6),(-6 - 2, 1 + 4)]`
= `[(5, 2),(6, 2),(-8, 5)]` ....(ii)
From (i) and (ii), we get
(A + B) + C = A + (B + C).
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