Permanents of matrices of signed ones

Ian M. Wanless*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

17 Citations (Scopus)


By calculating the permanents for all Hadamard matrices of orders up to and including 28 we answer a problem posed by E.T.H. Wang and a similar question asked by H. Perfect. Both questions are answered by the existence of Hadamard matrices of order 20 which do not seem to be simply related but nevertheless have the same permanent. For orders up to and including 20 we also settle several other existence questions involving permanents of (+1, -1)-matrices. Specifically, we establish the lowest positive value taken by the permanent in these cases and find matrices which have equal permanent and determinant when such a matrix exists. Our results address Conjectures 19 and 36 and Problems 5 and 7 in Mine's well known catalogue of unsolved problems on permanents. We also include a little-known proof that there exists a (+1,-1)-matrix A of order n such that per(A) = 0 if and only if n+ 1 is not a power of 2.

Original languageEnglish
Pages (from-to)427-433
Number of pages7
JournalLinear and Multilinear Algebra
Issue number6
Publication statusPublished - Nov 2005
Externally publishedYes


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