Covariance matrix

If X is a random vector
 ,
the covariance matrix is




Det($\Sigma$) = $\prod_{i=1}^n\lambda_i \geq 0$ where $\lambda_i$'s are eigenvalues of $\Sigma$.

Since $\Sigma$ is symmetric and positive definite, it can be diagonalized and its eigenvalues are all real and positive and the eigenvectors are orthogonal.
$$\begin{align} det(\Sigma) = det(V\Lambda V^T) = det(V)\cdot det(\Lambda) \cdot det(V^T) = det(\Lambda) \end{align}$$

$det(V) = \pm 1$ because $det(VV^{-1}) = det(V)det(V^{-1}) = det(V)det(V^T) = det(V)^2 = 1$

References:
http://www.ece.unm.edu/faculty/bsanthan/EECE-541/covar.pdf