I am wondering if there exist any square symmetric matrices $A$ with integer entries, all zeros along the diagonal, determinant $1$, and the property that none of the entries in the matrix are equal to $\pm$ 1. I noticed that they are not any of size 2 or 3.
Unimodular symmetric integral matrices with diagonal 0 and no $\pm 1$ entries
bilinear-formintegerslinear algebramatrices
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Best Answer
Try $$ \left[ \begin {array}{cccc} 0&5&2&4\\ 5&0&3&3 \\ 2&3&0&5\\ 4&3&5&0\end {array} \right]$$