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Question
Given that A = `[(1, 3),(5, -1)]`, B = `[(1, -1, 2),(3, 5, 2)]`, C = `[(1, 3, 2),(-4, 1, 3)]` verify that A(B + C) = AB + AC
Solution
Given A = `[(1, 3),(5, -1)]`, B = `[(1, -1, 2),(3, 5, 2)]` C = `[(1, 3, 2),(-4, 1, 3)]`
B + C = `[(1, -1, 2),(3, 5, 2)] + [(1, 3, 2),(-4, 1, 3)]`
= `[(2, 2, 4),(-1, 6, 5)]`
A(B + C) = `[(1, 3),(5, -1)] xx [(2, 2, 4),(-1, 6, 5)]`
= `[(2 - 3, 2 + 18, 4 + 15),(10 + 1, 10 - 6, 20 - 5)]`
= `[(-1, 20, 19),(11, 4, 15)]` ...(1)
AB = `[(1, 3),(5, -1)] xx [(1, -1, 2),(3, 5, 2)]`
= `[(1 + 9, -1 + 15, 2 + 6),(5 - 3, -5 - 5, 10 - 2)]`
= `[(10, 14, 8),(2, -10, 8)]`
AC = `[(1, 3),(5, -1)] xx [(1, 3, 2),(-4, 1, 3)]`
= `[(1 - 12, 3 + 3, 2 + 9),(5 + 4, 15 - 1, 10 - 3)]`
= `[(-11, 6, 11),(9, 14, 7)]`
AB + AC = `[(10, 14, 8),(2, -10, 8)] + [(-11, 6, 11),(9, 14, 7)]`
= `[(-1, 20, 19),(11, 4, 15)]` ...(2)
From (1) and (2) we get
A(B + C) = AB + AC
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