Sign matrices with |per A| = |det A|

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Minc’s list · Problem 7

Sign matrices with |per A| = |det A|

Open

Posed by
Wang
First posed
1974

Statement

Posed by E. T. H. Wang in 1974; it is Problem 7 in Minc’s 1978 book, reworded by Minc in 1987.

Problem 7. For what values of \(n\) do there exist nonsingular \(n\times n\) \((1,-1)\)-matrices \(A\) such that \(\lvert\operatorname{per}A\rvert=\lvert\det A\rvert\)?

Wang originally asked whether such matrices exist for every \(n\ge4\), which fails already at \(n=4\); Minc reworded the question in 1987. Asking for \(\operatorname{per}A=\det A\) gives the same set of orders, since swapping two rows keeps the permanent and changes the sign of the determinant.

Prior progress

  • There is no solution for \(n=2,3,4\) or for \(n=2^k-1\) with \(k\ge2\), and there are solutions for \(n=5,6\) (Kräuter and Seifter 1983, as reported by Minc 1987).
  • Wanless (2005) gave explicit solutions for every \(n\) from \(8\) to \(20\) except \(15\), so the answer was known for all \(n\le20\).

Our progress so far

  • Orders \(21\) to \(25\). We constructed explicit nonsingular \((\pm1)\)-matrices of orders \(21,22,23,24,25\) with \(\operatorname{per}A=-\det A\); for example the common absolute value at order \(21\) is \(11\cdot2^{22}\). So for \(n\le25\) solutions exist exactly when \(n\in\{1,5,6,8,\dots,14,16,\dots,25\}\), and \(n=26\) is the smallest open order.

References

  1. E. T. H. Wang, On permanents of (1, −1)-matrices, Israel J. Math. 18 (1974), 353–361. doi:10.1007/BF02760844
  2. A. R. Kräuter and N. Seifter, On some questions concerning permanents of (1, −1)-matrices, Israel J. Math. 45 (1983), 53–62. doi:10.1007/BF02760670
  3. H. Minc, Theory of permanents 1982–1985, Linear and Multilinear Algebra 21 (1987), 109–148. doi:10.1080/03081088708817786
  4. I. M. Wanless, Permanents of matrices of signed ones, Linear and Multilinear Algebra 53 (2005), 427–433. doi:10.1080/03081080500093990