Remarks on generalized magic squares of order 3. (English) Zbl 1160.05310

Păltineanu, Gavriil (ed.) et al., Trends and challenges in applied mathematics. Conference proceedings of the international conference, ICTCAM 2007, Bucharest, Romania, June 20–23, 2007. Bucharest: Matrix Rom (ISBN 978-973-755-283-9/pbk). 173-176 (2007).
Summary: A generalized magic square of order \(n\) is an \(n\) by \(n\) array of numbers whose rows, columns, and the two diagonals sum to \(k\), called the magic sum. In the first part of this paper, we give several elementary properties of generalized magic squares of order 3. In the second part we prove that any generalized magic square of order 3 is a \(^{-t}EP\) element if and only if the magic sum \(S^*\) of the adjoint matrix \(A^*\) is a non-zero number. As a consequence, the group inverse of generalized magic square is also a generalized-magic square, when the adjoint matrix has the magic sum non-zero.
For the entire collection see [Zbl 1135.00011].


05B15 Orthogonal arrays, Latin squares, Room squares
15A09 Theory of matrix inversion and generalized inverses
15A18 Eigenvalues, singular values, and eigenvectors