Determinants of box products of paths

Suppose that GG is the graph obtained by taking the box product of a path of length nn and a path of length mm. Let M be the adjacency matrix of GG. In 1996, Rara showed that, if n=mn=m, then det(M)=0. We extend this result to allow nn and mm to be any positive integers, and show that det(M)={0if gcd(n+1,m+1)≠1,(−1)nm/2if gcd(n+1,m+1)=1.