Coavarianc matrix determinant and inverse

Question

Let $C$ be a positive semi-definite symmetric covariance matrix; $I$ be an identity matrix; $\sigma^{2}$ a constant

How can I find the determinant and inverse of $$|C+\sigma^{2}I|$$ and $$(C+\sigma^{2}I)^{-1}$$

I am mostly interested in a method to separate $C$ from $\sigma^{2}$ or simplify the calculations

Answer 1

You must compute the matrix $D = C + \sigma^2 I$ first and then compute the inverse or determinant of $D$. It cannot be done faster for a general symmetric positive semi-definite matrices $C$.