Two miniatures relating the 2 × 2 minors of a square matrix and the symmetric group
DOI:
https://doi.org/10.17398/Keywords:
determinant, rank, minor, symmetric group, variant of magic squareAbstract
In the first miniature we recall a puzzle proposed by Martin Gardner in 1957 and refreshed by Blasco in 2014. We prove some properties of the matrices involved. We show that Gardner magical matrices are tropically singular. In the second miniature we construct a counterexample to the following conjecture by R. Flores: for a square matrix A = (aij ) of size n, if for each permutation σ there exists a 2 × 2 minor of A such that
aσ(k)k aσ(l)l - aσ(k)l aσ(l)k
vanishes, then determinant of A vanishes.
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[1] F. Blasco, El universo matemágico de Martin Gardner. Un cuarto de siglo de matemáticas recreativas, Investigación y Ciencia, Temas 77, 3er. trimestre, (2014), p. 7.
[2] M. Gardner, A new kind of magic square with remarkable properties, Scientific American 196 (1) (1957), 138 – 142.
[3] J. Matoušek, “Thirty–three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra”, Student Mathematical Library, Vol. 53, AMS, 2010.
[4] M.J. de la Puente, Isocanted cube: the lower Lebesgue volumes, incidence numbers and symmetries of this d–dimensional tile, Extracta Math. 40 (1) (2025), 1 – 26.
[5] I. Seco Serra, M. del M. Gómez Talavera, M. Aguado Molina, Amuletos planetarios e hispanohebreos bajomedievales y renacentistas del Museo Arqueológico Nacional, Boletı́n del Museo Arqueológico Nacional, n. 21–22–23, (2003–2004–2005) (Madrid).
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