Question

If `"A" = [(1 , 2),(-2 , 3)], "B" = [(2 , 1),(2 , 3)] "C" = [(-3 , 1),(2 , 0)]` verify that
(AB)C = A(BC),

Answer 1

AB = `[(1 , 2),(-2 , 3)] [(2 , 1),(2 , 3)]`

= `[(2 + 4, 1+6),(-4 + 6 , -2 + 9)] = [(6 , 7),(2 , 7)]`

(AB)C = `[(6 , 7),(2 , 7)][(-3 , 1),(2 , 0)]`

= `[(-18 + 14 , 6 + 0),(-6 + 14, 2 +0)] = [(-4 , 6),(8 , 2)]`

Now , BC = `[(2 , 1),(2 , 3)][(-3 , 1),(2 , 0)]`

= `[(-6 + 2 , 2 + 0),(-6 + 6 , 2 + 0)] = [(-4 , 2),(0 , 2)]`

A(BC) = `[(1 , 2),(-2 , 3)][(-4 , 2),(0 , 2)]`

= `[(-4 + 0 , 2 + 4),(8 + 0 , -4 + 6)] = [(-4 , 6),(8 , 2)]`
Hence, (AB)C = A(BC).