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$A$ is a matrix sized $n\times n$, all elements in $A$ are $\pm1$. Find $\max |A|$.

My Attempt
Denote $f(n)=\max |A_{n\times n}|$.
$f(1)=1$.
$f(2)=2$ is also obviously.
If $n\ge2$, $|A|$ must be even.
For $n=3$, $f(3)\ge4$ because $\left|\begin{array} r1&1&1\\1&-1&1\\1&-1&-1 \end{array}\right|=4$. Also, $|A|=A_{11} A_{22} A_{33}+A_{12} A_{23} A_{31}+A_{13} A_{21} A_{32}-A_{13} A_{22} A_{31}-A_{11} A_{23} A_{32}-A_{12} A_{21} A_{33}$ can not be $6$ since $A_{11} A_{22} A_{33}A_{12} A_{23} A_{31}A_{13} A_{21} A_{32}A_{13} A_{22} A_{31}A_{11} A_{23} A_{32}A_{12} A_{21} A_{33}$ must be $1$. $1\ne 1\cdot1\cdot1\cdot(-1)\cdot(-1)\cdot(-1)$.
Hence we have $f(3)\ne 6$. $f(3)=4$.
For $n\ge4$, I have no idea where to start with $f(n)$.
A trivial bound is $0<f(n)\le n!$.
EDIT
Related question:Maximum value of Determinant of $3 \times 3$ Matrix with entries $\pm 1$
It is not duplicated since I am discussing $n\times n$ determinants, not $3\times3$.

Kemono Chen
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    $f(n) \leq n^{2}$ because any eigen value $\lambda$ is bounded by $n$ and $|A|$ is the product of the eigen values. – Kavi Rama Murthy Sep 10 '18 at 08:36
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    Related: oeis A003433 – Kemono Chen Sep 10 '18 at 09:01
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    This problem is called "Hadamard maximal determinant problem". – Kemono Chen Sep 10 '18 at 09:17
  • Thanks for the information. I am not much of an expert on matrix theory; the bound $n!$ looked too crude, so I gave an obvious, better bound. – Kavi Rama Murthy Sep 10 '18 at 09:22
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    Your question seems to be an open problem... Anyway, consider adding the first column to each of the others. This does not change the determinant, but since $-1\equiv 1 \pmod 2$, each column $c_i$ can now be written as $2c_i'$ where $c_i'$ has integer entries. Thus $|A|=2^{n-1}|A'|$, where $|A'|$ is an integer. – Gabriel Romon Sep 10 '18 at 09:32
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    There's also a Hadamard bound for the determinant of a matrix. Look it up. – Gerry Myerson Sep 10 '18 at 10:21
  • Mathoverflow:https://mathoverflow.net/questions/39786/whats-the-maximum-determinant-of-the-0-1-matrix-from-mn-r/39824#39824 Mathworld reference:http://mathworld.wolfram.com/HadamardsMaximumDeterminantProblem.html – Kemono Chen Sep 11 '18 at 05:46
  • My recollection seems to be WOM (=write once memory). I was sure $n=428$ is the smallest open case of whether a Hadamard matrix exists (relevant to this question only when $n$ is a multiple of four), but Mathworld says $428$ was settled in the affirmative in 2004, and $n=668$ is the now the smallest open case. – Jyrki Lahtonen Sep 15 '18 at 19:53
  • Anyway, see an old answer of mine for the answer when $n=12$ (Paley construction IIRC). Neil Sloane has a webpage with more information. – Jyrki Lahtonen Sep 15 '18 at 19:56
  • I had some years ago followed a new path, using "varimin"-rotation known from principal-components-analysis in statistics. See my related question at https://mathoverflow.net/questions/90558/for-an-approach-to-the-hadamard-matrix-problem-is-there-a-proof-that-the-itera . Unfortunately I couldn't push my ansatz towards a definite solution... – Gottfried Helms Sep 17 '18 at 20:39

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From comments:

This is a semi-well-known open problem known as Hadamard's maximum determinant problem.

hmakholm left over Monica
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