A 
-matrix is a matrix whose elements
consist only of the numbers 
, 0, or 1. The number of distinct 
-
matrices (counting row and column permutations, the
transpose, and multiplication by 
as equivalent) having 
different row and column sums for 
, 4, 6, ... are 1, 4, 39, 2260, 1338614, ... (OEIS A049475).
For example, the 
matrix is given by
![[-1 -1; 0 1].](/images/equations/-101-Matrix/NumberedEquation1.svg) |
To get the total number from these counts (assuming that 0 is not the missing sum, which is true for 
),
multiply by 
.
In general, if an 
-matrix which has 
different column and row sums (collectively called line sums;
Bodendiek and Burosch 1995), then
1. 
is even.
2. The number in 
that does not appear as a line sum is either 
or 
.
3. Of the 
largest line sums, half are column sums and half are row sums.
For an 
-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 
, 2, .... The numbers of 
-matrices having maximum determinants are 1, 4, 240,
384, 30720, ... (OEIS A051753).
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