Let F(α) = [(cosα -sinα 0), (sinα cosα 0), (0 0 1)] and G(β) [(cosβ 0 sinβ), (0 1 0), (-sinβ 0 cosβ)]. Show that [F(α)G(β)]^–1 = G – ( – β) F( – α).

Question

Let \(F(α)=\begin{bmatrix}cosα&-sinα&0\\sinα&cosα&0\\0&0&1\end{bmatrix}\) and\(G(β)\begin{bmatrix}cosβ&0&sinβ\\0&1&0\\-sinβ&0&cosβ\end{bmatrix}\). 

Show that [F(α)G(β)]^–1 = G – ( – β) F( – α).

Answer 1

We have to show that

[F(α)G(β)] ^–1 = G( – β) F( – α)

We have already shown that

[G (β)] ^–1 = G( – β)

[F (α)] ^–1 = F( – α)

And LHS = [F(α)G(β)] ^–1

= [G (β)] ^–1 [F (α)] ^–1

= G( – β) F( – α)

Hence = RHS