I wonder how to express the determinant of a block covariance matrix. For example, I have a covariance matrix

$\Sigma=\left[ \begin{array}{cc} \Sigma_1 & \Sigma_{12} \\ \Sigma_{21} & \Sigma_2 \\ \end{array} \right]$

where $\Sigma_1\in\mathbb{R}^{n_1}\times\mathbb{R}^{n_1}$, $\Sigma_2\in\mathbb{R}^{n_2}\times\mathbb{R}^{n_2}$ and hence $\Sigma_{12}=\Sigma_{21}^\intercal\in\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}$.

Can I express the denterminant of $\Sigma$ in terms of the 3 blocks? That is:

$|\Sigma|=f(\Sigma_1,\Sigma_2,\Sigma_{12})$

I guess it is somehow related to Schur Complement, I am still studying..