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

Question

Let  \(F(\alpha)=\begin{bmatrix}cos\alpha&-sin\alpha&0\\sin\alpha&cos\alpha&0\\0&0&1\end{bmatrix}\) and \(G(\beta)=\begin{bmatrix}cos\beta&0&sin\beta\\0&1&0\\-sin\beta&0&cos\beta\end{bmatrix}\). 

Show that [F (α)] ^–1 = F( – α)]

Answer 1

\(F(\alpha)=\begin{bmatrix}cos\alpha&-sin\alpha&0\\sin\alpha&cos\alpha&0\\0&0&1\end{bmatrix}\)

| F(α)| = \(cos^2\alpha+sin^2\alpha\) = 1

Cofactors of A are:

C₁₁ = cos α C₂₁ = sin α C₃₁ = 0

C₁₂ = – sin α C₂₂ = cos α C₃₂ = 0

C₁₃ = 0 C₂₃ = – 10 C₃₃ = 1

Hence, [F (α)] ^–1 = F( – α)