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What is the number of non-singular $n \times n$ matrices with entries in $\{1, -1\}$? Here I assume that two such matrices are equivalent if one can be obtained from the other by permutations of rows and columns, or change of signs of rows and columns. An estimate would help.

How about non-singular matrices with entries in $\{1, 0, -1\}$?

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  • $\begingroup$ What do you get for $n=1, n=2, n=3$? $\endgroup$
    – Gerald Edgar
    Commented Jan 30, 2022 at 21:07
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    $\begingroup$ this can be equivalently phrased as probability a random $\pm 1$ $n\times n$ matrix is non-singular, which is very hard (I think) $\endgroup$
    – mathworker21
    Commented Jan 30, 2022 at 22:35
  • $\begingroup$ @Gerald only one. $\endgroup$
    – Arun
    Commented Jan 31, 2022 at 6:18
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    $\begingroup$ With respect to @mathworker21's comment, this recent pre-print tightens some old bounds, but the old bounds are enough to say that asymptotically almost all such matrices are non-singular. $\endgroup$
    – Peter Taylor
    Commented Jan 31, 2022 at 11:11
  • $\begingroup$ @Peter Thank you very much. Please post your comment as an answer, I will upvote it. $\endgroup$
    – Arun
    Commented Jan 31, 2022 at 11:22

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