A (-1,1)-matrix is a matrix whose elements consist only of the numbers -1 or 1. For an n×n (-1,1)-matrix, the largest possible determinants (Hadamard's maximum determinant problem) for n=1, 2, ... are 1, 2, 4, 16, 48, 160, ... (OEIS A003433; Ehrlich and Zeller 1962, Ehrlich 1964), the same as for an n×n (-1,0,1)-matrix. The numbers of distinct n×n (-1,1)-matrices having the largest possible determinant are 1, 4, 96, 384, 30720, ... (OEIS A188895).

See also

Hadamard Matrix, Integer Matrix

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Ehrlich, H. "Determinantenabschätzungen für binäre Matrizen." Math. Z. 83, 123-132, 1964.Ehrlich, H. and Zeller, K. "Binäre Matrizen." Z. angew. Math. Mechanik 42, T20-21, 1962.Kahn, J.; Komlós, J.; and Szemeredi, E. "On the Probability that a Random +/-1 Matrix is Singular." J. Amer. Math. Soc. 8, 223-240, 1995.Seifter, N. "Upper Bounds for Permanents of (1,-1)-Matrices." Israel J. Math. 48, 69-78, 1984.Sloane, N. J. A. Sequences A003433/M1291 and A188895 in "The On-Line Encyclopedia of Integer Sequences."Wang, E. T.-H. "On Permanents of (1,-1)-Matrices." Israel J. Math. 18, 353-361, 1974.

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Cite this as:

Weisstein, Eric W. "(-1,1)-Matrix." From MathWorld--A Wolfram Web Resource.

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