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)]`

Answer 1

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