The square matrix formed by setting , where is a th root of unity. The Schur matrix has a particularly simple determinant given by
(1)
where is an odd prime and
(2)
This determinant has been used to prove the quadratic reciprocity law (Landau 1958, Vardi 1991). The absolute values of the permanents of the Schur matrix of order are given by 1, 3, 5, 105, 81, 6765, ... (Sloane's A003112, Vardi 1991). Denote the Schur matrix with the first row and first column omitted by . Then
square matrix formed by setting 
, where 
is a 
th root of unity. The Schur matrix has a particularly simple determinant given by 
is an odd prime and 
are given by 1, 3, 5, 105, 81, 6765, ... (Sloane's A003112, Vardi 1991).
with the first row and first column omitted by 
. Then