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Question
If A = `[(1, 2),(-2, 1)]`, B = `[(2, 3),(3, -4)]` and C = `[(1, 0),(-1, 0)]`, verify: A(B + C) = AB + AC
Solution
We have, A = `[(1, 2),(-2, 1)]`, B = `[(2, 3),(3, -4)]` and C = `[(1, 0),(-1, 0)]`
B + C = `[(2, 3),(3, -4)] + [(1, 0),(-1, 0)]`
= `[(3, 3),(2, -4)]`
⇒ A · (B + C) = `[(1, 2),(-2, 1)] [(3, 3),(2, -4)]`
= `[(3 + 4, 3 - 8),(-6 + 2, -6 - 4)]`
= `[(7, -5),(-4, -10)]` .....(iii)
AB = `[(1, 2),(-2, 1)] [(2, 3),(3, -4)]`
= `[(2 + 6, 3 - 8),(-4 + 3, -6 - 4)]`
= `[(8, -5),(-1, -10)]`
And AC = `[(1, 2),(-2, 1)] [(1, 0),(-1, 0)]`
= `[(1 - 2, 0),(-2 - 1, 0)]`
= `[(-1, 0),(-3, 0)]`
∴ AB + AC = `[(8, -5),(-1, -10)] + [(-1, 0),(-3, 0)]`
= `[(7, -5),(-4, -10)]` ......(iv)
From equations (iii) and (iv), we get
A · (B + C) = A · B+ A · C
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