A055165
Number of invertible n X n matrices with entries equal to 0 or 1.
Original entry on oeis.org
1, 1, 6, 174, 22560, 12514320, 28836612000, 270345669985440, 10160459763342013440
Offset: 0
Ulrich Hermisson (uhermiss(AT)server1.rz.uni-leipzig.de), Jun 18 2000
For n=2 the 6 matrices are {{{0, 1}, {1, 0}}, {{0, 1}, {1, 1}}, {{1, 0}, {0, 1}}, {{1, 0}, {1, 1}}, {{1, 1}, {0, 1}}, {{1, 1}, {1, 0}}}.
- Eric Weisstein's World of Mathematics, Nonsingular Matrix.
- Chai Wah Wu, Can machine learning identify interesting mathematics? An exploration using empirically observed laws, arXiv:1805.07431 [cs.LG], 2018.
- Miodrag Zivkovic, Classification of small (0,1) matrices, arXiv:math/0511636 [math.CO], 2005; Linear Algebra and its Applications, 414 (2006), 310-346.
- Miodrag Zivkovic, Classification of (0,1) matrices of order not exceeding 8.
- Index entries for sequences related to binary matrices
A046747(n) + a(n) = 2^(n^2) = total number of n X n (0, 1) matrices = sequence
A002416.
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a(n)=sum(t=0,2^n^2-1,!!matdet(matrix(n,n,i,j,(t>>(i*n+j-n-1))%2))) \\ Charles R Greathouse IV, Feb 09 2016
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from itertools import product
from sympy import Matrix
def A055165(n): return sum(1 for s in product([0,1],repeat=n**2) if Matrix(n,n,s).det() != 0) # Chai Wah Wu, Sep 24 2021
More terms from Miodrag Zivkovic (ezivkovm(AT)matf.bg.ac.rs), Feb 28 2006
A046747
Number of n X n rational {0,1}-matrices of determinant 0.
Original entry on oeis.org
1, 10, 338, 42976, 21040112, 39882864736, 292604283435872, 8286284310367538176
Offset: 1
Günter M. Ziegler (ziegler(AT)math.tu-berlin.de)
a(2)=10: the matrix of all 0's, 4 matrices with 2 zeros in the same row or column, 4 matrices with 3 zeros and the all-1 matrix.
- J. Bourgain, V. Vu and P. M. Wood, On the Singularity Probability of Discrete Random Matrices, Journal of Functional Analysis, 258 (2010), 559-603.
- R. P. Brent and J. H. Osborn, Bounds on minors of binary matrices, arXiv preprint arXiv:1208.3330 [math.CO], 2012.
- J. Kahn, J. Komlos and E. Szemeredi, On the probability that a random ±1-matrix is singular, J. AMS 8 (1995), 223-240.
- J. Komlos, On the determinant of (0,1)-matrices, Studia Math. Hungarica 2 (1967), 7-21.
- N. Metropolis and P. R. Stein, On a class of (0,1) matrices with vanishing determinants, J. Combin. Theory, 3 (1967), 191-198.
- Eric Weisstein's World of Mathematics, Singular Matrix.
- Miodrag Zivkovic, Classification of small (0,1) matrices, arXiv:math/0511636 [math.CO], 2005.
- Miodrag Zivkovic, Classification of small (0,1) matrices, Linear Algebra and its Applications, 414 (2006), 310-346.
- Index entries for sequences related to binary matrices
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Sum[KroneckerDelta[Det[Array[Mod[Floor[k/(2^(n*(#1-1)+#2-1))],2]&,{n,n}]],0],{k,0,(2^(n^2))-1}] (* John M. Campbell, Jun 24 2011 *)
Count[Det /@ Tuples[{0, 1}, {n, n}], 0] (* David Trimas, Sep 23 2024 *)
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A046747(n) = m=matrix(n,n); ct=0; for(x=0,2^(n*n)-1,a=binary(x+2^(n*n)); for(i=1,n, for(j=1,n,m[i,j]=a[n*i+j+1-n])); if(matdet(m)==0,ct=ct+1,); ); ct \\ Randall L Rathbun
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a(n)=sum(i=0,2^n^2-1,matdet(matrix(n,n,x,y,(i>>(n*x+y-n-1))%2))==0) \\ Charles R Greathouse IV, Feb 21 2015
A056990
Number of nonsingular n X n (-1,1)-matrices.
Original entry on oeis.org
1, 2, 8, 192, 22272, 11550720, 25629327360, 236229525504000, 8858686914082897920, 1331751782100764385607680
Offset: 0
a(5) from Winston C. Yang (winston(AT)cs.wisc.edu), Aug 26 2000
A118992
Number of real n X n invertible symmetric (+1,-1) matrices.
Original entry on oeis.org
2, 4, 32, 512, 16896, 1190144, 163899904, 46195853312, 25585116626944, 28281621931343872
Offset: 1
A057981
Number of singular n X n (-1,0,1)-matrices.
Original entry on oeis.org
0, 1, 33, 7875, 15099201, 237634987683, 30805715676309201
Offset: 0
a(0)-a(5) confirmed and a(6) added by
Minfeng Wang, May 01 2024
A197487
Number of nonsingular n X n matrices with elements from {0,1,2}.
Original entry on oeis.org
1, 2, 50, 12792, 30844560, 671869521960, 129553882116606720
Offset: 0
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(* 2x2 case *) cnt = 0; Do[d = Det[{{a, b}, {c, d}}]; If[d != 0, cnt++], {a, 0, 2}, {b, 0, 2}, {c, 0, 2}, {d, 0, 2}]; cnt (* T. D. Noe, Nov 29 2011 *)
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