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)

Answer 1

(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).