Given a base on a vector space of dimension n, we can represent a tensor of order r with a hypermatrix of dimension n and order r. Then, the standard determinant tensor is represented by a hypermatrix H of order and dimension n. Gherardelli showed that the Cayley determinant of H, times n!, is equal to the number of even Latin squares of order n minus the number of odd Latin squares of order n. The Alon᎐Tarsi conjecture says that this difference is not zero, whenever n is even. If n is odd the difference is zero, but the conjecture can be extended to the odd case by computing the difference only for Latin squares which have the entries of the diagonal equal to 1. In this paper we use the Laplace rule in order to compute the Cayley determinant, and we prove that the difference between the number of even Latin squares and the number of odd Latin squares is nonnegative. We also prove the Alon᎐Tarsi conjecture for Latin squares of order c2 r , where r is a positive integer and either c is an even integer for which the Alon-Tarsi conjecture is true, or c is an odd integer such that the extended Alon-Tarsi conjecture is true for c and for c q 1. ᮊ 1997 Academic Press n n if all the elements of the diagonal are equal to 1.