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Question
Verify that A(B + C) = AB + AC of the following matrix:
A = `[(1, -1, 3),(2, 3, 2)], "B" = [(1, 0),(-2, 3),(4, 3)], "C" = [(1, 2),(-2, 0),(4, -3)]`
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Solution
A(B + C) = `[(1, -1, 3),(2, 3, 2)] {[(1, 0),(-2, 3),(4, 3)] + [(1, 2),(-2, 0),(4, -3)]}`
= `[(1, -1, 3),(2, 3, 2)] [(1 + 1, 0 + 2),(-2 - 2, 3 + 0),(4 + 4, 3 - 3)]`
= `[(1, -1, 3),(2, 3, 2)][(2, 2),(-4, 3),(8, 0)]`
= `[(2 + 4 + 24, 2 - 3 + 0),(4 - 12 + 16, 4 + 9 + 0)]`
= `[(30, -1),(8, 13)]` ...(i)
`"AB" + "AC" = [(1, -1, 3),(2, 3, 2)] [(1, 0),(-2, 3),(4, 3)] + [(1, -1, 3),(2, 3, 2)] [(1, 2),(-2, 0),(4, -3)]`
`= [(1 + 2 + 12, 0 - 3 + 9),(2 - 6 + 8, 0 + 9 + 6)] + [(1 + 2 + 12, 2 + 0 - 9),(2 - 6 + 8, 4 + 0 - 6)]`
= `[(15, 6),(4, 15)] + [(15, -7),(4, -2)]`
= `[(15 + 15, 6 - 7),(4 + 4, 15 - 2)]`
= `[(30, -1),(8, 13)]` ...(ii)
From (i) and (ii), we get
A(B + C) = AB + AC
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