Let $\bf{X}\sim N_p(\mu,\Sigma)$ with $|\bf{\Sigma}|\neq 0$. Check that $$|\bf{\Sigma}|=|\bf{\Sigma}_{22}||\bf{\Sigma}_{11}-\bf{\Sigma}_{12}\bf{\Sigma}_{22}^{-1}\bf{\Sigma}_{21}|$$ (Hint: Note that $|\bf{\Sigma}|$ can be factored into the product of contributions from the marginal and conditional distributions.)
First thing I noticed is that I should write $\bf{\Sigma}$ as block matrix so $$\bf{\Sigma}=\begin{matrix}| \Sigma_{11}& \Sigma_{12}|\\ |\Sigma_{21} & \Sigma_{22}|\end{matrix}$$ (not write the bold). I made some research here and fount it Proofs of Determinants of Block matrices , so if $\Sigma_{11}$ is invertible I could get the result, but there is no guarantee about it. Anyone can give me some light about the hint given, not understood it.
