suppose $P_n$ and $P_m$ are paths with $n$ and $m$ edges respectively.consider $A_n$ and $A_m$ as adjacency matrix of them.now I want to calculate the number of perfect matching of $P_n \square P_m$ (it is cartesian product http://en.wikipedia.org/wiki/Cartesian_product_of_graphs )
now I consider $A_{m,n}=A_m \otimes I_n + I_m \otimes A_n$ which is adjacency matrix for $P_n \square P_m$. ($\otimes$ is Kronecker product http://en.wikipedia.org/wiki/Kronecker_product)
now because $P_n \square P_m$ is bipartite graph then the number of perfect matching is equal to $\sqrt{per(A_{m,n})}$.
now I know that if I define $A^{*}_{m,n}=A_m \otimes I_n + iI_m \otimes A_n$ which $i^2=-1$ it is enough to show that $ det(A^{*}_{m,n})=per(A_{m,n})$
my problem is to show this later equation which is so great relation.
please give me the way to prove that,any help will be great,thanks.