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  "body": "\n## Problem\n\nLet $n \\ge 1$ be an integer, and let $\\det$ denote the [[determinant|00KDZRW4C0]].\nFind the largest value of $\\det A$ over all $n \\times n$ matrices $A$ whose entries are all $1$ or $-1$.\n\n## Remarks\n\n[[Hadamard's inequality|CGDN1X32NQ]] gives $\\abs{\\det A} \\le n^{n/2}$ for such matrices.\nEquality holds exactly when the rows of $A$ are pairwise orthogonal, that is, when $A$ is a Hadamard matrix.\nA Hadamard matrix of order $n$ can exist only for $n = 1$, $n = 2$ or $n$ divisible by $4$.\nThe Hadamard conjecture, that one exists for every multiple of $4$, is open.\nFor other orders the maximum is smaller than $n^{n/2}$, and it is known only for some values of $n$.\n\nThe determinant of an $n \\times n$ matrix with entries $\\pm 1$ is divisible by $2^{n-1}$.\nThe maximum for $n$ equals $2^{n-1}$ times the maximal determinant of an $(n-1) \\times (n-1)$ matrix with entries $0$ and $1$.\nSo the problem is the same as the maximal determinant problem for $0/1$ matrices.\n\nSmall cases: for $n = 2$, the maximum is $2$, reached by $\\begin{pmatrix} 1 \u0026 1 \\\\ 1 \u0026 -1 \\end{pmatrix}$.\nFor $n = 3$, the bound $3^{3/2} \\approx 5.2$ and the divisibility by $4$ show that the maximum is at most $4$, and $4$ is reached.\n\n## Explanations\n\n### L1 human\n\nFill a square grid with the numbers $1$ and $-1$.\nThe problem asks how large its determinant can be.\nGeometrically, it asks how large the volume of a box can be when its edges join the centre of a cube to corners of the cube.\nThe best case is when all edges are at right angles, but this is possible only for some sizes.\n",
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      "first_stated": "1893",
      "posed_by": "Jacques Hadamard"
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    "name": "Hadamard's maximal determinant problem",
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      "wikipedia": "Hadamard's maximal determinant problem"
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  "msc": [
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  "name": "Hadamard's maximal determinant problem",
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