Variance of correlated noise
Even though your calculation yields the correct result (in this case), the steps are not completely correct.

First of all, the covariance matrix of a random variable $n$ is given by 

$$ C = E[nn^H]. $$

The nomenclature of variance is more suitable for scalar random variables, in my opinion (or it refers to the diagonal of the covariance matrix, i.e. the variance of each element of $n$). 

Now, you state the covariance of $n$ should be $\sigma^2A$. Hence, we have

$$ E[nn^H] = \sigma^2A $$.

Now, you are interested in the covariance matrix of the random variable $y=A^{-1}n$, which is given by 

$$
D = E[A^{-1}nn^HA^{-H}]
$$
Now, knowing that $A$ is constant, we can take it out of the expectation, to get

$$
D = A^{-1}E[nn^H]A^{-H}=A^{-1}\sigma^2A A^{-H}=\sigma^2A^{-H}.
$$

Now, knowing that $A$ is a covariance matrix and these are generally semi-positive definite and symmetric, we have $A^{-1}=A^{-H}$, so the final result is

$$
D = \sigma^2 A^{-1}
$$

which matches your guess. Note that in case you multiply $n$ with a different matrix $B$, the result is not $\sigma^2B^2A$ (as you might have guessed), but according to above calculation it is $\sigma^2BAB^H$.

I often refer to the following link, when it comes to expectations that involve matrices: http://www.psi.toronto.edu/matrix/expect.html