  We prove that for any $n\times n$ matrix, $A$, and $z$ with $|z|\geq \|A\|$,
we have that $\|(z-A)^{-1}\|\leq\cot (\frac{\pi}{4n}) \dist (z,
\spec(A))^{-1}$. We apply this result to the study of random orthogonal
polynomials on the unit circle.
