If $\mathrm{A}=\left(\begin{array}{ccc}\mathrm{cos\alpha }& \mathrm{sin\alpha }& 0\\ -\mathrm{sin\alpha }& \mathrm{cos\alpha }& 0\\ 0& 0& 1\end{array}\right) \mathrm{then} {\mathrm{AA}}^{\mathrm{T}}={\mathrm{A}}^{\mathrm{T}}\mathrm{A}=$

$$\begin{gathered} {\mathrm{AA}}^{\mathrm{T}}=\left[\begin{array}{ccc}\cos \mathrm{\alpha }& \sin \mathrm{\alpha }& 0\\ -\sin \mathrm{\alpha }& \cos \mathrm{\alpha }& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos \mathrm{\alpha }& -\sin \mathrm{\alpha }& 0\\ \sin \mathrm{\alpha }& \cos \mathrm{\alpha }& 0\\ 0& 0& 1\end{array}\right] \\ =\left[\begin{array}{ccc}{\cos }^2\mathrm{\alpha }+{\sin }^2\mathrm{\alpha }+0& \cancel{-\sin \mathrm{\alpha } \cos \mathrm{\alpha }}+\cancel{\sin \mathrm{\alpha } \cos \mathrm{\alpha }+0}& 0\\ -\sin \mathrm{\alpha } \cos \mathrm{\alpha }+\cos \mathrm{\alpha } \sin \mathrm{\alpha }+0& {\sin }^2\mathrm{\alpha } \mathrm{t} {\cos }^2\mathrm{\alpha }+0& 0\\ 0& 0& 1\end{array}\right] \\ =\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right] \\ =\mathrm{I} \\ \mathrm{Similarly}, {\mathrm{A}}^{\mathrm{T}}\mathrm{A}=\mathrm{I} \end{gathered}$$