$\left|\begin{array}{ccc}\cos \mathrm{\theta }& -\sin \mathrm{\theta }& 0\\ \sin \mathrm{\theta }& \cos \mathrm{\theta }& 0\\ 0& 0& 1\end{array}\right|$

$\Rightarrow \left|\begin{array}{ccccccc}\cos \mathrm{\theta }& -\sin \mathrm{\theta }& 0& ⋮& 1& 0& 0\\ \sin \mathrm{\theta }& \cos \mathrm{\theta }& 0& ⋮& 0& 1& 0\\ 0& 0& 1& ⋮& 0& 0& 1\end{array}\right|$

⇒  R₂ – R₂ (cosθ) – R₁ (sinθ)

$\Rightarrow \left|\begin{array}{ccccccc}\cos \mathrm{\theta }& -\sin \mathrm{\theta }& 0& ⋮& 1& 0& 0\\ 0& {\cos }^2\mathrm{\theta }+{\sin }^2\mathrm{\theta }& 0& ⋮& -\mathrm{sin\theta }& 1& 0\\ 0& 0& 1& ⋮& 0& 0& 1\end{array}\right|$

⇒  R₁ – R₁ + R₂ sinθ

$\Rightarrow \left|\begin{array}{ccccccc}\cos \mathrm{\theta }& 0& 0& ⋮& 1-{\sin }^2\mathrm{\theta }& \mathrm{sin\theta }& 0\\ 0& 1& 0& ⋮& -\mathrm{sin\theta }& 1& 0\\ 0& 0& 1& ⋮& 0& 0& 1\end{array}\right|$

$\Rightarrow \left|\begin{array}{ccccccc}\cos \mathrm{\theta }& 0& 0& ⋮& {\cos }^2\mathrm{\theta }& \mathrm{sin\theta }& 0\\ 0& 1& 0& ⋮& -\mathrm{sin\theta }& 1& 0\\ 0& 0& 1& ⋮& 0& 0& 1\end{array}\right|$

$\Rightarrow \left|\begin{array}{ccccccc}1& 0& 0& ⋮& \mathrm{cos\theta }& \mathrm{sin\theta }& 0\\ 0& 1& 0& ⋮& -\mathrm{sin\theta }& \mathrm{cos\theta }& 0\\ 0& 0& 1& ⋮& 0& 0& 1\end{array}\right| {\mathrm{R}}_1-{\mathrm{R}}_1/\mathrm{cos\theta }$

$\therefore {\mathrm{A}}^{-1}=\left(\begin{array}{ccc}\cos \mathrm{\theta }& \sin \mathrm{\theta }& 0\\ -\sin \mathrm{\theta }& \cos \mathrm{\theta }& 0\\ 0& 0& 1\end{array}\right)$