# Any trick to swap order of determinant and matrix inverse operation?

Been thinking through fitting a kind of Gaussian mixture model in more of a neural network style (kind of similar to RNADE or RMADE by Larochelle, without going into details) and see that this could scale really nicely to very large dimensions if I could avoid having to know both $\Sigma$ AND $\Sigma^{-1}$ per the multivariate Gaussian equation. I.e., from what I'm seeing, the Gaussian PDF...

$det(2\pi\Sigma)^{-\frac12}e^{-\frac12(x-\mu)^t \Sigma^{-1}(x-\mu)}$

seems to require both the covariance and precision matrices. I bet the answer is no, but is there a way I could fit parameters for the precision matrix and avoid having to invert it to get the normalization constant (e.g., if i could swap the order of the determinant and inverse sqrt operation, and just need the $\Sigma^{-\frac12}$, I could do a Cholesky decomposition on the precision matrix).

• Are you looking for $det(A^{-1})=det(A)^{-1}$? Feb 1, 2018 at 7:50
• ya so that is a legit property? Like the transpose of the inverse is the inverse of the transpose?
– JPJ
Feb 1, 2018 at 20:02
• Well yes. More generally det(AB) = det(A) det(B). It's very surprising that you know very advanced stuff like Cholesky decomposition but you have doubts about this fact! Feb 1, 2018 at 20:10
• Well....guess this is the first time I've needed to know it :-)
– JPJ
Feb 1, 2018 at 20:28