Multivariate normal distributions density function: What is the value if the determinant is zero? Given a function that calculates a multivariate normal distribution, what value should be returned if the determinant of the covariance matrix is zero? What does it mean if this is the case?
Edit (I hope this makes it clearer):
I have data that is assumed to be described by a MVN distribution.
 
Now I want to sum over this the results of the data to get the likelihood.
 A: Return value for your function
If you want to write code for a function that computes a multivariate normal distribution, then, in case the determinant of the var-covar is zero, this return-value is undefined because the var-covar matrix is singular (the inverse does not exist). 
Probably the best return value for your function, in case the var-covar matrix is singular, is NA. 
Reasons for singular var-covar matrix
The reasons for singularity could be that one of your 'variables' is (as good as) constant such that its variance is (almost) zero, or it could also be that one of your variables is a linear combination of the other variables. 
A: Presume that the covariance matrix as specified is n by n, i.e., the Multivariate Normal random variable X is in n dimensions.  Det(covariance matrix) = 0 if and only if the covariance matrix is singular.  If the covariance matrix is singular, X does not have a density.  There may exist a lower dimensional space (manifold) in which X is concentrated, such that the covariance matrix in that lower dimensional space is nonsingular, in which case X, when projected into that lower dimensional space, would have a density.
Edit: Pertaining to whuber's comment above, based on the thread title, I was presuming the OP wants to know (how to compute) the density.  The cumulative probability distribution does exist, even if the covariance matrix is singular (has determinant = 0), and could be computed by integrating the lower dimensional density.  I believe the only way that there could not be a lower dimensional manifold on which X was concentrated and which had nonsingular covariance, is if X were concentrated at a single point (for example, a one dimensional Normal having variance = 0), in whuch case the cumulative distribution = 1 for any region containing that point, and 0 otherwise.
A: Actually it can also be defined by the use of generalized inverse and "pseudo-determinant". 
Solution is not unique though.
