Covariance of Inverse of Random Matrix Is there a way to calculate the Covariance of the inverse of a random matrix from the Covariance of the original matrix? So say you have C1 = Cov(X), is there a direct way to calculate C2 = Cov(inv(X))?

No. You can't even calculate $E[1/X]$ from $E[X]$ in general for a single random variable, let alone the (co)variance of it.
Example: let $X$ be a Rademacher RV, so its variance is $1$ and the variance of $1/X$ is also $1$ (because $1/X=X$ for this particular distribution).
And, let $Y$ be a RV with possible values $2$, and $-2$, i.e. a slight variation of the Rademacher RV, and let $P(Y=2)=p$. So, the variance of $Y$ is
$$\operatorname{var}(Y)=E[Y^2]-E[Y]^2=4-(2p-2(1-p))^2$$
And, this is equal to $1$ if $p=(2+\sqrt 3)/4$. Now, calculating $\operatorname{var}(1/Y)$:
$$\operatorname{var}(1/Y)=E[1/Y^2]-E[1/Y]^2=1/4-(p/2-(1-p)/2)^2\neq 1$$
when calculated. So, both random variables have the same variance but not the same variance for their reciprocals, which means you can't actually find $\operatorname{var}(1/Y)$ from $\operatorname{var}(Y)$.
