$\begin{aligned} & \text { Given, } A(\alpha, \beta)=\left[\begin{array}{rcc}\cos \alpha & \sin \alpha & 0 \\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & e^\beta\end{array}\right] \\ & \text { Now, }|A(\alpha, \beta)|=e^\beta\left(\cos ^2 \alpha+\sin ^2 \alpha\right)=e^\beta \\ & \qquad C_{11}=e^\beta \cos \alpha, C_{12}=e^\beta \sin \alpha, C_{13}=0 \\ & C_{21}=-e^\beta \sin \alpha, C_{22}=e^\beta \cos \alpha, C_{23}=0 \\ & C_{31}=0, C_{32}=0, C_{33}=\cos ^2 \alpha+\sin ^2 \alpha=1 \\ & \therefore \operatorname{adj}(A(\alpha, \beta))=\left[\begin{array}{ccc}e^\beta \cos \alpha & -e^\beta \sin \alpha & 0 \\ e^\beta \sin \alpha & -e^\beta \cos \alpha & 0 \\ 0 & 0 & 1\end{array}\right] \\ & \therefore[A(\alpha, \beta)]^{-1}=\frac{1}{e^\beta} \times e^\beta\left[\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & e^{-\beta}\end{array}\right] \\ & \quad=\left[\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & e^{-\beta}\end{array}\right]=A(-\alpha,-\beta)\end{aligned}$