I know that for any given matrix $U\in\mathbb{R}^{3\times 3}$ the following matrix $$ B = cI_3 + dU^TU $$ is invertible, where $I_3$ is the three by three identity matrix and $c,d>0$ are constants coming from the modelled physical system. Precisely, in my case I have that $U=ST$ where $S,T\in\mathbb{R}^{3\times 3}$ are both skew-symmetric matrices. I even know that if the matrix $U^TU$ has rank 1, then there is an iterative way to compute this inverse.
Is there some reference or some formula I can see which applies to explicit the inversion of the matrix $B$? If it may help, I know that the matrix $U$ has rank 2 and it is singular.
If it may help someone, I have inverted it with the Cayley-Hamilton Theorem, since the eigenvalues in my case are pretty easy to compute.
So, I get an expression similar to this one $$ B^{-1} = c_1B^2 + c_2B + c_3I_3 $$ which is something quite simple to study.