  We show that a perturbation of any fixed square matrix D by a random unitary
matrix is well invertible with high probability. A similar result holds for
perturbations by random orthogonal matrices; the only notable exception is when
D is close to orthogonal. As an application, these results completely eliminate
a hard-to-check condition from the Single Ring Theorem by Guionnet, Krishnapur
and Zeitouni.
