I have a given Matrix equation
$R(s)^{'}_{3\times 3} = \psi(s)_{3\times 3}R(s)\tag 1$
Conditions
- R(s) is orthogonal and determinent 1. Can say in the format of rotation matrix
- $R^{'}(s)$ derivative w.r.t s,has determinent $0$ . So it cant have inverse
Question
- What I need to do is to switch the terms of $R^{'}(s),R(s)$ between L.H.S and R.H.S using their inverse. More precisely I need to make $R(s)^{-1}$ alone on L.H.S. But the issue is $\left({\frac{\mathrm{d}R(s) }{\mathrm{d} s}} \right )^{-1}$ doest not have an inverse so I am struggling to take that to R.H.S to make a form I desire. How do we do that? Is there anyway to make this $\left({\frac{\mathrm{d}R(s) }{\mathrm{d} s}} \right )$ to a determinent non zero matrix by addition?