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Question
Verify the property A(B + C) = AB + AC, when the matrices A, B, and C are given by A = `[(2, 0, -3),(1, 4, 5)]`, B = `[(3, 1),(-1, 0),(4, 2)]` and C = `[(4, 7),(2, 1),(1,-1)]`
Solution
A = `[(2, 0, -3),(1, 4, 5)]`
B = `[(3, 1),(-1, 0),(4, 2)]`
C = `[(4, 7),(2, 1),(1,-1)]`
B + C = `[(3, 1),(-1, 0),(4, 2)] + [(4, 7),(2, 1),(1, -1)]`
= `[(3 + 4, 1 + 7),(-1 + 2, 0 + 1),(4 + 1, 2 - 1)]`
B + C =`[(7, 8),(1, 1),(5, 1)]`
A(B + C) = `[(2, 0, -3),(1, 4, 5)] [(7, 8),(1, 1),(5, 1)]`
= `[(14 + 0 - 15, 16 0- 3),(7 + 4 + 25, 8 + 4 + 5)]`
A(B + C) = `[(-1, 13),(36, 17)]` ......(1)
AB = `[(2, 0, -3),(1, 4, 5)] [(3, 1),(-1, 0),(4, 2)]`
= `[(6 + 0 - 12, 2 + 0 - 6),(3 - 4 + 20, 1 + 0 + 10)]`
AB = `[(-6, -4),(19, 11)]`
AC = `[(2, 0, -3),(1, 4, 5)] [(4, 7),(2, 1),(1, -1)]`
= `[(8 + 0 - 3, 14 + 0 + 3),(4 + 8 + 5, 7 + 4 - 5)]`
AC = `[(5, 17),(17, 6)]`
AB + AC = `[(-6, -4),(19, 11)] +[(5, 17),(17, 6)]`
= `[(-6 +5,-4 + 17),(19 + 17, 11 + 6)]`
AB + AC = `[(-1, 13),(36, 17)]` ......(2)
From equation (1) and (2)
A(B + C) = AB + AC
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