A (-1,0,1)-matrix is a matrix whose elements consist only of the numbers -1, 0, or 1. The number of distinct (-1,0,1)-n×n matrices (counting row and column permutations, the transpose, and multiplication by -1 as equivalent) having 2n different row and column sums for n=2, 4, 6, ... are 1, 4, 39, 2260, 1338614, ... (OEIS A049475). For example, the 2×2 matrix is given by

 [-1 -1;  0  1].

To get the total number from these counts (assuming that 0 is not the missing sum, which is true for n<=10), multiply by (2n!)^2. In general, if an n×n (-1,0,1)-matrix which has 2n different column and row sums (collectively called line sums; Bodendiek and Burosch 1995), then

1. n is even.

2. The number in {-n,1-n,2-n,...,n} that does not appear as a line sum is either -n or n.

3. Of the n largest line sums, half are column sums and half are row sums.

For an n×n (-1,0,1)-matrix, the largest possible determinants (Hadamard's maximum determinant problem) are the same as for a (-1,1)-matrix, i.e., 1, 2, 4, 16, 48, 160, ... (OEIS A003433; Ehrlich 1964, Brenner and Cummings 1972) for n=1, 2, .... The numbers of n×n (-1,0,1)-matrices having maximum determinants are 1, 4, 240, 384, 30720, ... (OEIS A051753).

See also

Alternating Sign Matrix, C-Matrix, Integer Matrix

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Bodendiek, R. and Burosch, G. "Solution to the Antimagic 0,1,-1 Matrix Problem." Aufgabe 5.30 in Streifzüge durch die Kombinatorik: Aufgaben und Lösungen aus dem Schatz der Mathematik-Olympiaden. Heidelberg, Germany: Spektrum Akademischer Verlag, pp. 250-253, 1995.Brenner, J. and Cummings, L. "The Hadamard Maximum Determinant Problem." Amer. Math. Monthly 79, 626-630, 1972.Ehrlich, H. "Determinantenabschätzungen für binäre Matrizen." Math. Z. 83, 123-132, 1964.Sloane, N. J. A. Sequences A003433/M1291, A049475, and A051753 in "The On-Line Encyclopedia of Integer Sequences."

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Cite this as:

Weisstein, Eric W. "(-1,0,1)-Matrix." From MathWorld--A Wolfram Web Resource.

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