  Let k be an arbitrary field (of arbitrary characteristic) and let X =
[x_{i,j}] be a generic m x n matrix of variables. Denote by I_2(X) the ideal in
k[X] = k[x_{i,j}: i = 1, ..., m; j = 1, ..., n] generated by the 2 x 2 minors
of X. We give a recursive formulation for the lengths of the k[X]-module
k[X]/(I_2(X) + (x_{1,1}^q,..., x_{m,n}^q)) as q varies over all positive
integers using Grobner basis. This is a generalized Hilbert-Kunz function, and
our formulation proves that it is a polynomial function in q. We give closed
forms for the cases when m is at most 2, %as well as the closed forms for some
other special length functions. We apply our method to give closed forms for
these Hilbert-Kunz functions for cases $m \le 2$.
