<s>[INST] Does a matrix $A$ of $\pm1$'s of order $11$ exist with $\det A >4000$? How to prove or disprove that statement: there exists a square matrix of size 11 having all entries $ \pm 1$ and its determinant greater than  4000?
 [/INST] I found such matrix, having executed a few times the Maple code $$interface(rtablesize = 15):with(LinearAlgebra):with(Statistics): $$ $$P := Array([-1, 1]):X := RandomVariable(EmpiricalDistribution(P)): $$ $$M := convert(Matrix(11, 11, convert(Sample(X, 121), list)),rational); Determinant(M);$$ which outputed $$ \left[ \begin {array}{ccccccccccc} - 1&- 1& 1&- 1& 1&- 1&
 1& 1& 1& 1& 1\\- 1& 1&- 1&- 1&-
 1& 1&- 1& 1& 1& 1& 1\\ - 1& 1&
 1& 1&- 1& 1& 1& 1&- 1&- 1&- 1\\ 
 1& 1& 1&- 1& 1& 1& 1&- 1&- 1& 1& 1
\\  1&- 1&- 1& 1& 1&- 1&- 1&- 1& 1
& 1& 1\\  1&- 1& 1& 1&- 1&- 1&- 1&
- 1& 1& 1& 1\\  1-& 1& 1&- 1&- 1&-
 1&- 1& 1&- 1& 1& 1\\  1&- 1& 1&
 1& 1&- 1& 1&- 1&- 1& 1&- 1\\ - 1&
- 1& 1&- 1&- 1& 1&- 1& 1& 1& 1&- 1
\\  1&- 1& 1&- 1& 1&- 1& 1& 1&- 1&
- 1&- 1\\  1& 1&- 1& 1& 1&- 1&- 1&
- 1& 1&- 1&- 1\end {array} \right] 
 $$ and $4096. $
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