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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 = `[(2, -3),(5, -4),(-6, 1)] + [(-1, 2),(2, 2), (0, 3)]`
= `[(2 + (-1), -3 + 2),(5 + 2, -4 + 2),(-6 + 0, 1 + 3)]`
= `[(1, -1),(7, -2),(-6, 4)]`
∴ (A + B) + C = `[(1, -1),(7, -2),(-6, 4)] + [(4, 3),(-1, 4),(-2, 1)]`
= `[(1 + 4, -1 + 3),(7 + (-1), -2 + 4),(-6 + (-2), 4 + 1)]`
= `[(5, 2),(6, 2),(-8, 5)]` ...(1)
Also, B + C = `[(-1, 2),(2, 2),(0, 3)] + [(4, 3),(-1, 4),(-2, 1)]`
= `[(-1 + 4, 2 + 3),(2 + (-1), 2 + 4),(0 + (-2), 3 + 1)]`
= `[(3, 5),(1, 6),(-2, 4)]`
∴ A + (B + C) = `[(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)]` ...(2)
From (1) and (2), we get,
(A + B) + C =A + (B + C).
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