Question

If A = `[(1, 2),(-2, 1)]`, B = `[(2, 3),(3, -4)]` and C = `[(1, 0),(-1, 0)]`, verify: A(B + C) = AB + AC

Answer 1

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