Solve this

Question:

Let $F(\alpha)=\left[\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{array}\right]$ and $G(\beta)=\left[\begin{array}{ccc}\cos \beta & 0 & \sin \beta \\ 0 & 1 & 0 \\ -\sin \beta & 0 & \cos \beta\end{array}\right]$

(i) $[F(\alpha)]^{-1}=F(-\alpha)$

(ii) $[G(\beta)]^{-1}=G(-\beta)$

(iii) $[F(\alpha) G(\beta)]^{-1}=G(-\beta) F(-\alpha)$.

Solution:

$(\mathrm{i}) F(\alpha)=\left[\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{array}\right]$

$\Rightarrow F(-\alpha)=\left[\begin{array}{ccc}\cos (-\alpha) & -\sin (-\alpha) & 0 \\ \sin (-\alpha) & \cos (-\alpha) & 0 \\ 0 & 0 & 1\end{array}\right]=\left[\begin{array}{ccc}\cos \alpha & \sin \alpha & 0 \\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{array}\right]$

Now,

$C_{11}=\left|\begin{array}{cc}\cos \alpha & 0 \\ 0 & 1\end{array}\right|=\cos \alpha, C_{12}=-\left|\begin{array}{cc}\sin \alpha & 0 \\ 0 & 1\end{array}\right|=-\sin \alpha$ and $C_{13}=\left|\begin{array}{cc}\sin \alpha & \cos \alpha \\ 0 & 0\end{array}\right|=0$

$C_{21}=-\left|\begin{array}{cc}-\sin \alpha & 0 \\ 0 & 1\end{array}\right|=\sin \alpha, C_{22}=\left|\begin{array}{cc}\cos \alpha & 0 \\ 0 & 1\end{array}\right|=\cos \alpha$ and $C_{23}=-\left|\begin{array}{cc}\cos \alpha & -\sin \alpha \\ 0 & 0\end{array}\right|=0$

$C_{31}=\left|\begin{array}{cc}-\sin \alpha & 0 \\ \cos \alpha & 0\end{array}\right|=0, C_{32}=-\left|\begin{array}{cc}\cos \alpha & 0 \\ \sin \alpha & 0\end{array}\right|=0$ and $C_{33}=\left|\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right|=1$

$\Rightarrow \operatorname{adj}\{F(\alpha)\}=\left[\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{array}\right]^{T}=\left[\begin{array}{ccc}\cos \alpha & \sin \alpha & 0 \\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{array}\right]$

$\Rightarrow|F(\alpha)|=1$

$\therefore[F(\alpha)]^{-1}=\left[\begin{array}{ccc}\cos \alpha & \sin \alpha & 0 \\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{array}\right]$     ....(1)

$\Rightarrow[F(\alpha)]^{-1}=F(-\alpha)$

$($ ii $) G(\beta)=\left[\begin{array}{ccc}\cos \beta & 0 & \sin \beta \\ 0 & 1 & 0 \\ -\sin \beta & 0 & \cos \beta\end{array}\right]$

$\Rightarrow G(-\beta)=\left[\begin{array}{ccc}\cos (-\beta) & 0 & \sin (-\beta) \\ 0 & 1 & 0 \\ -\sin (-\beta) & 0 & \cos (-\beta)\end{array}\right]=\left[\begin{array}{ccc}\cos \beta & 0 & -\sin \beta \\ 0 & 1 & 0 \\ \sin \beta & 0 & \cos \beta\end{array}\right]$

Now,

$C_{11}=\left|\begin{array}{cc}1 & 0 \\ 0 & \cos \beta\end{array}\right|=\cos \beta, C_{12}=-\left|\begin{array}{cc}0 & 0 \\ -\sin \beta & \cos \beta\end{array}\right|=0$ and $C_{13}=\left|\begin{array}{cc}0 & 1 \\ -\sin \beta & 0\end{array}\right|=\sin \beta$

$C_{21}=-\left|\begin{array}{cc}0 & \sin \beta \\ 0 & \cos \beta\end{array}\right|=0, C_{22}=\left|\begin{array}{cc}\cos \beta & \sin \beta \\ -\sin \beta & \cos \beta\end{array}\right|=1$ and $C_{23}=-\left|\begin{array}{cc}\cos \beta & 0 \\ -\sin \beta & 0\end{array}\right|=0$

$C_{31}=\left|\begin{array}{cc}0 & \sin \beta \\ 1 & 0\end{array}\right|=-\sin \beta, C_{32}=-\left|\begin{array}{cc}\cos \beta & \sin \beta \\ 0 & 0\end{array}\right|=0$ and $C_{33}=\left|\begin{array}{cc}\cos \beta & 0 \\ 0 & 1\end{array}\right|=\cos \beta$

$\operatorname{adj}\{G(\beta)\}=\left[\begin{array}{ccc}\cos \beta & 0 & \sin \beta \\ 0 & 1 & 0 \\ -\sin \beta & 0 & \cos \beta\end{array}\right]^{T}=\left[\begin{array}{ccc}\cos \beta & 0 & -\sin \beta \\ 0 & 1 & 0 \\ \sin \beta & 0 & \cos \beta\end{array}\right]$

$|G(\beta)|=1$

$\therefore G(\beta)^{-1}=\left[\begin{array}{ccc}\cos \beta & 0 & -\sin \beta \\ 0 & 1 & 0 \\ \sin \beta & 0 & \cos \beta\end{array}\right]$      ....(2)

$\Rightarrow G(\beta)^{-1}==G(-\beta)$

$($ iii $) F(\alpha)=\left[\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{array}\right]$

$\Rightarrow F(-\alpha)=\left[\begin{array}{ccc}\cos (-\alpha) & -\sin (-\alpha) & 0 \\ \sin (-\alpha) & \cos (-\alpha) & 0 \\ 0 & 0 & 1\end{array}\right]=\left[\begin{array}{ccc}\cos \alpha & \sin \alpha & 0 \\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{array}\right]$         ....(3)

$G(\beta)=\left[\begin{array}{ccc}\cos \beta & 0 & \sin \beta \\ 0 & 1 & 0 \\ -\sin \beta & 0 & \cos \beta\end{array}\right]$

$\Rightarrow G(-\beta)=\left[\begin{array}{ccc}\cos (-\beta) & 0 & \sin (-\beta) \\ 0 & 1 & 0 \\ -\sin (-\beta) & 0 & \cos (-\beta)\end{array}\right]=\left[\begin{array}{ccc}\cos \beta & 0 & -\sin \beta \\ 0 & 1 & 0 \\ \sin \beta & 0 & \cos \beta\end{array}\right]$      ...(4)

$[F(\alpha) G(\beta)]^{-1}=[G(\beta)]^{-1}[F(\alpha)]^{-1}$

$=\left[\begin{array}{ccc}\cos \beta & 0 & -\sin \beta \\ 0 & 1 & 0 \\ \sin \beta & 0 & \cos \beta\end{array}\right]\left[\begin{array}{ccc}\cos \alpha & \sin \alpha & 0 \\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{array}\right]$    [Using eqn (1) and (2)]

$=G(-\beta) F(-\alpha)$        [Using eqn (3) and (4)]