Question

If A = `[(2, 3),(1, 4)], B = [(1, 0),(0, 2)], C = [(2, 0),(-1, 1)]`, find

1. A(BC)
2. (AB)C
3. If A(BC) = (AB)C

Answer 1

(i)

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

= `[(1 xx 2 + 0 xx (-1), 1 xx 0 + 0 xx 1),(0 xx 2 + 2 xx (-1), 0 xx 0 + 2 xx 1)]`

= `[(2, 0),(-2, 2)]`

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

= `[(2 xx 2 + 3 xx (-2), 2 xx 0 + 3 xx 2),(1 xx 2 + 4 xx (-2), 1 xx 0 + 4 xx 2)]`

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

= `[(-2, 6),(-6, 8)]`

(ii)

AB = `[(2, 3),(1, 4)] xx [(1, 0),(0, 2)]`

= `[(2 xx 1 + 3 xx 0, 2 xx 0 + 3 xx 2),(1 xx 1 + 4 xx 0, 1 xx 0 + 4 xx 2)]`

= `[(2 + 0, 0 + 6),(1 + 0, 0 + 8)]`

= `[(2, 6),(1, 8)]`

(AB)C = `[(2, 6),(1, 8)] xx [(2, 0),(-1, 1)]`

= `[(2 xx 2 + 6 xx (-1), 2 xx 0 + 6 xx 1),(1 xx 2 + 8 xx (-1), 1 xx 0 + 8 xx 1)]`

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

= `[(-2, 6),(-6, 8)]`

(iii)

A(BC) = `[(-2, 6),(-6, 8)]`

(AB)C = `[(-2, 6),(-6, 8)]`

∴ A(BC) = (AB)C