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Question
Verify that A(B + C) = AB + AC of the following matrix:
A = `[(4, -2),(2, 3)], "B" = [(-1, 1),(3, -2)] and "C" = [(4, 1),(2, -1)]`
Solution
A(B + C) = `[(4, -2),(2, 3)] {[(-1, 1),(3, -2)] + [(4, 1),(2, -1)]}`
= `[(4, -2),(2, 3)] [(-1 + 4, 1 + 1),(3 + 2, -2-1)]`
= `[(4, -2),(2, 3)][(3, 2),(5, -3)]`
= `[(12 - 10, 8 + 6),(6 + 15, 4 - 9)]`
= `[(2, 14),(21, -5)]` ...(i)
AB + AC = `[(4, -2),(2, 3)] [(-1, 1),(3, -2)] + [(4, -2),(2, 3)] [(4, 1),(2, -1)]`
= `[(-4 - 6, 4 + 4),(-2 + 9, 2 - 6)] + [(16 - 4, 4 + 2),(8 + 6, 2 - 3)]`
= `[(-10, 8),(7, -4)] + [(12, 6),(14, -1)]`
= `[(-10 + 12, 8 + 6),(7 + 14, -4 - 1)]`
= `[(2, 14),(21, -5)]` ...(ii)
From (i) and (ii), we get
A(B + C) = AB + AC
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