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  Volume 10, Issue 3, Article 63
An Upper Bound for the Determinant of a Matrix with given Entry Sum and Square Sum

    Authors: Ortwin Gasper, Hugo Pfoertner, Markus Sigg,  
    Keywords: Determinant, Matrix Inequality, Hadamard's Determinant Theorem, Hadamard Matrix.  
    Date Received: 05/03/2009  
    Date Accepted: 15/09/2009  
    Subject Codes:

15A15, 15A45, 26D07.

    Editors: Sever S. Dragomir,  

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 $ ertdet Mert le ertlphaert(eta-delta)^{(n-1)/2}$ for real $ n 	imes n$-matrices $ M$, where $ n lpha$ and $ n eta$ are the sum of the entries and the sum of the squared entries of $ M$, respectively, and $ delta := (lpha^2-eta)/(n-1)$, provided that $ lpha^2 ge eta$. This result is applied to find an upper bound for the determinant of a matrix whose entries are a permutation of an arithmetic progression.

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