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Question
If A = `[(1, 3),(2, 4)]`, B = `[(1, 2),(4, 3)]` and C = `[(4, 3),(1, 2)]`, find:
- (AB)C
- A(BC)
Is A(BC) = (AB)C?
Solution
i (AB)C = `[(1, 3),(2, 4)][(1, 2),(4, 3)] [(4, 3),(1, 2)]`
= `[(1(1) + 3(4), 1(2) + 3(3)),(2(1) + 4(4), 2(2) + 4(3))] = [(4, 3), (1, 2)]`
= `[(13, 11),(18, 16)][(4, 3),(1, 2)]`
= `[(13(4) + 11(1), 13(3) + 11(2)),(18(4) + 16(1), 18(3) + 16(2))]`
= `[(52 + 11, 39 + 22),(72 + 16, 54 + 32)]`
= `[(63, 61),(88, 86)]`
ii A(BC) = `[(1, 3),(2, 4)][(1, 2),(4, 3)][(4, 3),(1, 2)]`
= `[(1, 3),(2, 4)] [(1(4) + 2(1), 1(3) + 2(2)),(4(4) + 3(1), 4(3) + 3(2))]`
= `[(1, 3),(2, 4)][(4 + 2, 3 + 4),(16 + 3, 12 + 6)]`
= `[(1, 3),(2, 4)][(6, 7),(19, 18)]`
= `[(6 + 57, 7 + 54),(12 + 76,14 + 72)]`
= `[(63, 61),(88, 86)]`
∴ A(BC) = (AB)C
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