Question

Let A = `[(1, 0),(2, 1)]`, B = `[(2, 3),(-1, 0)]`, Find A² + AB + B².

Answer 1

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

A² = A × A = `[(1 , 0),(2 , 1)] xx [(1 , 0),(2 , 1)]`

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

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

= `[(1, 0),(4, 1)]`

AB = A × B = `[(1, 0),(2, 1)] xx [(2, 3),(-1, 0)]`

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

= `[(2 + 0, 3 + 0),(4 + (-1), 6 + 0)]`

= `[(2, 3),(3, 6)]`

B² = B × B = `[(2, 3),(-1, 0)] xx [(2, 3),(-1, 0)]`

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

= `[(4 - 3, 6 + 0),(-2 + 0, -3 + 0)]`

= `[(1, 6),(-2, -3)]`

∴ A² + AB + B² = `[(1, 0),(4, 1)] + [(2, 3),(3, 6)] + [(1, 6),(-2, -3)]`

= `[(1 + 2 + 1, 0 + 3 + 6),(4 + 3 - 2, 1 + 6 - 3)]`

= `[(4, 9),(5, 4)]`