TITLE: What does Determinant of Covariance Matrix give?
QUESTION [32 upvotes]: I am representing my 3d data in convariance matrix. I just want to know what the determinant of Convariance Matrix gives. If the determinant is positive, zero, negative, high positive, high negative. What does it mean or represent?
Thanks 
EDIT:
Covariance is being used to represent variance for 3d coordiantes that I have. If my covariance matrix A determinant is +100, and the other covariance matrix B determinant is +5. Which of these values show if the variance is more or not. Which value tells that datapoints are more dispersed. Which value shows that readings are further away from mean.

REPLY [6 votes]: The determinant of the covariance matrix is referred to as generalized variance by Wilks in 1932. Comparing the density of the univariate and multivariate normal, it is easy to see that $|\Sigma|$ plays a similar role to $\sigma^2$.
This has several interpretations (see for example Anderson 2003, Section 7.5)

A geometric interpretation: it is proportional to the volume of the ellipsoid $\left\{u \in \mathcal{R}^{k} \mid(u-\mu)^{\prime} \Sigma^{-1}(u-\mu)=c^{2}\right\}$
An entropy interpretation, as discussed by @tmp

Relation to generalized correlation?
If $|\Sigma|$ is the generalized variance, is there also a generalized correlation? Defining $\sqrt{1-\frac{|\Sigma|}{\sigma_1^2\cdot\ldots\cdot\sigma^2_N}}$, this is called sometimes the collective correlation coefficient. You can verify that for N=2, this is indeed the usual correlation coefficent: $\sqrt{1-\frac{\sigma_1^2\sigma_2^2-\rho \sigma_1^2\sigma_2^2}{\sigma_1^2\sigma^2_2}}=\sqrt{1-(1-\rho^2)}=\rho$
References
Anderson, T. W., An introduction to multivariate statistical analysis., Wiley Series in Probability and Statistics. Hoboken, NJ: Wiley (ISBN 0-471-36091-0/hbk). xx, 721 p. (2003). ZBL1039.62044.
Wilks, S. S., Certain generalizations in the analysis of variance., Biometrika 24, 471-494 (1932). ZBL58.1172.02.