$\begin{aligned} & A(\alpha, \beta)=\left[\begin{array}{ccc}\cos \alpha & \sin \alpha & 0 \ -\sin \alpha & \cos \alpha & 0 \ 0 & 0 & e^\beta\end{array} ight] \ & \Rightarrow[A(\alpha, \beta)]^{-1}=\end{aligned}$

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$\begin{aligned} & A(\alpha, \beta)=\left[\begin{array}{ccc}\cos \alpha & \sin \alpha & 0 \ -\sin \alpha & \cos \alpha & 0 \ 0 & 0 & e^\beta\end{array}ight] \ & \Rightarrow[A(\alpha, \beta)]^{-1}=\end{aligned}$

Aicharya · Accepted Answer

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

$\begin{aligned} & A(\alpha, \beta)=\left[\begin{array}{ccc}\cos \alpha & \sin \alpha & 0 \\ -\sin \alpha &…

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