The determinants of certain matrices arising from the Boolean lattice

Let BB be the Boolean lattice on an nn-set with B=⊔BiB=⊔Bi the rank decomposition. Let M(n,i)M(n,i) be the incidence matrix between BiBi and Bn−iBn−i. We obtain a recursive formula for the determinant of the matrix M(n,i)M(n,i).