As explained in intuition of determinant : we know that determinant is the how the volume is scaled when the matrix is regarded as a projection.

Now, if we narrow down the definition of the matrix here: assuming that it is a covariance matrix, i.e., positive semi definitive and symmetric, will there be any new characteristics/intuition of determinant.

As per Ben's comments:

In a 2D space, covariance matrix represents an ellipse, which captures the distribution of a vector of 2 random variables. In such a context, what does determinant say about these 2 variables?