I have a 2-parameter (tilt,pan) rotation computed as tilt followed by pan, i.e. two rotation matrices multiplied together:
$$R(t,p)=\begin{pmatrix} c_p & s_p s_t & s_p c_t \\ 0 & c_t & -s_t \\ -s_p & c_p s_t & c_p c_t \end{pmatrix}$$
where $c,s$ are shortcuts of cosine and sine of the pan/tilt angle.
Is it possible to find $t',p'$ such that $R(t',p')=R(t,p)^{-1}$ ?
In other words, do these special rotations form a subspace of all possible rotations? In other words, does any tilt-pan rotation has corresponding inverse tilt-pan rotation?
Because the inverse of a rotation is just its transpose, you'd need to be able to express the $(1, 2)$ element of your matrix as the $(2, 1)$ entry of some other tilt-pan matrix. But that $(2,1)$ entry is always zero. So for any tilt-pan matrix $M$ where $s_p s_t \ne 0$, there is no other tilt-pan matrix whose inverse is $M$.