When this hold: $\det(AA^{T})=0?$ I know if A is a column vector, the equality holds.  Any comments or suggestions would be greatly appreciated. 

The determinant of a square matrix is nonzero if and only if the matrix is invertible (or, in other terminology, nonsingular).
If $A$ is a generic matrix with real coefficients, not necessarily square, we know that $AA^T$ is invertible if and only if $A$ has a right inverse. So the condition for having $\det(AA^T)=0$ is that the matrix doesn't have a right inverse, which can also be stated in the form “the matrix $A$ has rank less than the number of its rows”.
