A rank computation problem related to the HK function of trinomial hypersurfaces

In this article, I provide a solution to a rank computation problem related to the computation of the Hilbert-Kunz function for any disjoint-term trinomial hypersurface, over any field of characteristic 2. This rank computation problem was posed in Upadhyay (arXiv:1204.5417). The formula for the Hilbert-Kunz function for disjoint-term trinomial hypersurfaces can be deduced from the result(s) of this article, modulo a lot of tedious notation.