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Question
Verify A(BC) = (AB)C of the following case:
A = `[(2, 4, 3),(-1, 3, 2)], "B" = [(2, -2),(3, 3),(-1, 1)], "C" = [(3, 1),(1, 3)]`
Solution
BC = `[(2, -2),(3, 3),(-1, 1)] [(3, 1),(1, 3)]`
= `[(6 - 2, 2 - 6),(9 + 3, 3 + 9),(-3 + 1, -1 + 3)]`
= `[(4, -4),(12, 12),(-2, 2)]`
∴ A(BC) = `[(2, 4, 3),(-1, 3, 2)] [(4, -4),(12, 12),(-2, 2)]`
= `[(8 + 48 - 6, -8 + 48 + 6),(-4 + 36 - 4, 4 + 36 + 4)]`
= `[(50, 46),(28, 44)]` ...(1)
AB = `[(2, 4, 3),(-1, 3, 2)] [(2, -2),(3, 3),(-1, 1)]`
= `[(4 + 12 - 3, -4 + 12 + 3),(-2 + 9 - 2, 2 + 9 + 2)]`
= `[(13, 11),(5, 13)]`
∴ (AB)C = `[(13, 11),(5, 13)] [(3, 1),(1, 3)]`
= `[(39 + 11, 13 + 33),(15 + 13, 5 + 39)]`
= `[(50, 46),(28, 44)]` ...(2)
From (1) and (2), A(BC) = (AB)C.
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