Question

If A_α = `[(cosα, sinα),(-sinα, cosα)]`, then which of following statement is TRUE?

Options

• A_α.A_β = A_αβ and (A_α)ⁿ = `[(cos^nα, sin^nα),(-sin^nα, cos^nα)]`

• A_α.A_β = A_αβ and (A_α)ⁿ = `[(cosnα, sinnα),(-sinnα, cosnα)]`

• A_α.A_β = A_α+β and (A_α)ⁿ = `[(cos^nα, sin^nα),(-sin^nα, cos^nα)]`

• A_α.A_β = A_α+β and (A_α)ⁿ = `[(cosnα, sinnα),(-sinnα, cosnα)]`

Answer 1

`bb(A_α.A_β = A_(α+β) and (A_α)^n = [(cosnα, sinnα),(-sinnα, cosnα)])`

Explanation:

A_α = `[(cosα, sinα),(-sinα, cosα)]`

A_α.A_β = `[(cosα, sinα),(-sinα, cosα)][(cosβ, sinβ),(-sinβ, cosβ)]`

= `[(cosαcosβ - sinαsinβ, cosαsinβ + sinαcosβ),(-sinαcosβ - cosαsinβ, -sinαsinβ + cosαcosβ)]`

= `[(cos(α + β), sin(a + β)),(-sin(α + β), cos(α + β))]`

= A_(α+β)

Also, (A_α)ⁿ = `[(cosnα, sinnα),(-sinnα, cosnα)]`