Abstract
Let Ω n denote the set of alln×n-(1,−1)-matrices. E.T.H. Wang has posed the following problem: For eachn≧4, can one always find nonsingularA∈Ω n such that |perA|=|detA| (*)? We present a solution forn≦6 and, more generally, we show that (*) does not hold ifn=2k−1,k≧2, even for singularA∈Ω n . Moreover, we prove that perA≠0 ifA∈Ω n ,n=2k−1, and we derive new results concerning the divisibility of the permanent in Ω n by powers of 2.
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Kräuter, A.R., Seifter, N. On some questions concerning permanents of (1,−1)-matrices. Israel J. Math. 45, 53–62 (1983). https://doi.org/10.1007/BF02760670
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DOI: https://doi.org/10.1007/BF02760670