An upper bound for the determinant of a matrix with given entry sum and square sum.

ABSTRACT. By deducing characterisations of the matrices which have maximal determinant in the set of matrices with given entry sum and square sum, we prove the inequality | det M | ≤ |α|(β − δ) (n−1)/2 for real n × n-matrices M, where nα and nβ are the sum of the entries and the sum of the squared entries of M, respectively, and δ: = (α 2 − β)/(n − 1), provided that α 2 ≥ β. This result is applied to find an upper bound for the determinant of a matrix whose entries are a permutation of an arithmetic progression.