  We give a generating function for the number of pairs of $n\times n$ matrices
$(A, B)$ over a finite field that are mutually annihilating, namely, $AB=BA=0$.
This generating function can be viewed as a singular analogue of a series
considered by Cohen and Lenstra. We show that this generating function has a
factorization that allows it to be meromorphically extended to the entire
complex plane. We also use it to count pairs of mutually annihilating nilpotent
matrices. This work is essentially a study of the motivic aspects about the
variety of modules over $\mathbb C[u,v]/(uv)$ as well as the moduli stack of
coherent sheaves over an algebraic curve with nodal singularities.
