行列式为平方数的平方元素矩阵

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平方元素矩阵，顾名思义，就是其元素全都为平方数的矩阵。
可以通过行列互易对角化的矩阵为平凡解。我们要求非平凡解。
进一步，可以拓宽数域到有理数的平方数。
不仅要求非平凡解，甚至可以要求全部元素非零，称为无零解。
二阶易解，三阶的呢？

1. Det[( {   {Subscript[a, 11], Subscript[a, 12], Subscript[a, 13]},   {Subscript[a, 21], Subscript[a, 22], Subscript[a, 23]},   {Subscript[a, 31], Subscript[a, 32], Subscript[a, 33]}  } )]

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iniogateln

主对角线元素为平方数的三角阵

Charlslots

1,1,1
1,4,1
1,1,4
结果是9.
其实只要其他的都固定，留一个元素作为未知数x，结果是x的线性函数，解一个二次同余方程就行了。

Sherryufas

所以lsr的方案可以扩展到任意奇数阶无零解

singqing

第二行前两个数选16,25,第三行开始主对角线为4,所有其它元素为1,对偶数阶都可行

AzertVab

1. SL[X_] := Piecewise[{   {Reverse[Transpose[Reverse[X]]],     First[Diagonal[X]] > Last[Diagonal[X]]},   {X, First[Diagonal[X]] < Last[Diagonal[X]]},   {Piecewise[{      {Reverse[Transpose[Reverse[X]]],        First[Diagonal[X, 1]] > Last[Diagonal[X, 1]]},      {X, First[Diagonal[X, 1]] <= Last[Diagonal[X, 1]]}}],     First[Diagonal[X]] == Last[Diagonal[X]]}   }]Llst = {{{1, 1, 1}, {1, 4, 1}, {1, 1, 4}}, {{1, 1, 1}, {1, 4, 4}, {1,      4, 16}}, {{1, 1, 1}, {1, 9, 1}, {1, 1, 9}}, {{1, 1, 1}, {1, 9,      25}, {1, 25, 81}}, {{1, 1, 1}, {1, 16, 4}, {1, 4, 4}}, {{1, 1,      1}, {1, 81, 25}, {1, 25, 9}}, {{1, 1, 4}, {1, 9, 16}, {4, 16,      36}}, {{1, 4, 1}, {4, 36, 16}, {1, 16, 9}}, {{1, 4, 4}, {4, 1,      4}, {4, 4, 1}}, {{1, 4, 4}, {4, 1, 4}, {4, 4, 4}}, {{1, 4,      4}, {4, 4, 4}, {4, 4, 1}}, {{1, 4, 4}, {4, 9, 16}, {4, 16,      9}}, {{1, 4, 4}, {4, 25, 4}, {4, 4, 36}}, {{1, 4, 4}, {4, 25,      16}, {4, 16, 25}}, {{1, 4, 4}, {4, 36, 4}, {4, 4, 25}}, {{1, 9,      9}, {9, 1, 9}, {9, 9, 16}}, {{1, 9, 9}, {9, 16, 9}, {9, 9,      1}}, {{1, 9, 9}, {9, 16, 49}, {9, 49, 64}}, {{1, 9, 9}, {9, 64,      49}, {9, 49, 16}}, {{4, 1, 1}, {1, 1, 1}, {1, 1, 4}}, {{4, 1,      1}, {1, 4, 1}, {1, 1, 1}}, {{4, 1, 4}, {1, 1, 1}, {4, 1,      16}}, {{4, 1, 4}, {1, 49, 25}, {4, 25, 16}}, {{4, 4, 1}, {4, 16,      1}, {1, 1, 1}}, {{4, 4, 1}, {4, 16, 25}, {1, 25, 49}}, {{4, 4,      4}, {4, 1, 4}, {4, 4, 1}}, {{4, 4, 4}, {4, 9, 1}, {4, 1,      9}}, {{4, 4, 4}, {4, 9, 16}, {4, 16, 36}}, {{4, 4, 4}, {4, 36,      16}, {4, 16, 9}}, {{4, 4, 16}, {4, 9, 25}, {16, 25, 81}}, {{4, 9,      9}, {9, 4, 36}, {9, 36, 4}}, {{4, 9, 36}, {9, 4, 9}, {36, 9,      4}}, {{4, 16, 4}, {16, 81, 25}, {4, 25, 9}}, {{4, 36, 9}, {36, 4,      9}, {9, 9, 4}}, {{9, 1, 1}, {1, 1, 1}, {1, 1, 9}}, {{9, 1,      1}, {1, 9, 1}, {1, 1, 1}}, {{9, 1, 4}, {1, 9, 4}, {4, 4,      4}}, {{9, 1, 16}, {1, 1, 4}, {16, 4, 36}}, {{9, 1, 25}, {1, 1,      1}, {25, 1, 81}}, {{9, 4, 1}, {4, 4, 4}, {1, 4, 9}}, {{9, 4,      16}, {4, 1, 4}, {16, 4, 9}}, {{9, 4, 16}, {4, 4, 4}, {16, 4,      36}}, {{9, 4, 25}, {4, 4, 16}, {25, 16, 81}}, {{9, 16, 1}, {16,      36, 4}, {1, 4, 1}}, {{9, 16, 4}, {16, 9, 4}, {4, 4, 1}}, {{9, 16,      4}, {16, 36, 4}, {4, 4, 4}}, {{9, 16, 16}, {16, 36, 16}, {16,      16, 49}}, {{9, 16, 16}, {16, 49, 16}, {16, 16, 36}}, {{9, 25,      1}, {25, 81, 1}, {1, 1, 1}}, {{9, 25, 4}, {25, 81, 16}, {4, 16,      4}}, {{16, 1, 4}, {1, 1, 1}, {4, 1, 4}}, {{16, 4, 1}, {4, 4,      1}, {1, 1, 1}}, {{16, 4, 25}, {4, 4, 1}, {25, 1, 49}}, {{16, 9,      9}, {9, 1, 9}, {9, 9, 1}}, {{16, 9, 49}, {9, 1, 9}, {49, 9,      64}}, {{16, 25, 4}, {25, 49, 1}, {4, 1, 4}}, {{16, 49, 9}, {49,      64, 9}, {9, 9, 1}}, {{25, 4, 4}, {4, 1, 4}, {4, 4, 36}}, {{25, 4,      4}, {4, 36, 4}, {4, 4, 1}}, {{25, 4, 16}, {4, 1, 4}, {16, 4,      25}}, {{25, 16, 4}, {16, 25, 4}, {4, 4, 1}}, {{36, 4, 4}, {4, 1,      4}, {4, 4, 25}}, {{36, 4, 4}, {4, 25, 4}, {4, 4, 1}}, {{36, 4,      16}, {4, 1, 1}, {16, 1, 9}}, {{36, 4, 16}, {4, 4, 4}, {16, 4,      9}}, {{36, 16, 4}, {16, 9, 1}, {4, 1, 1}}, {{36, 16, 4}, {16, 9,      4}, {4, 4, 4}}, {{36, 16, 16}, {16, 9, 16}, {16, 16, 49}}, {{36,      16, 16}, {16, 49, 16}, {16, 16, 9}}, {{49, 1, 25}, {1, 4,      4}, {25, 4, 16}}, {{49, 16, 16}, {16, 9, 16}, {16, 16,      36}}, {{49, 16, 16}, {16, 36, 16}, {16, 16, 9}}, {{49, 25,      1}, {25, 16, 4}, {1, 4, 4}}, {{64, 9, 49}, {9, 1, 9}, {49, 9,      16}}, {{64, 49, 9}, {49, 16, 9}, {9, 9, 1}}, {{81, 1, 25}, {1, 1,      1}, {25, 1, 9}}, {{81, 16, 25}, {16, 4, 4}, {25, 4, 9}}, {{81,      25, 1}, {25, 9, 1}, {1, 1, 1}}, {{81, 25, 16}, {25, 9, 4}, {16,      4, 4}}};MatrixForm /@ Llst;MatrixForm /@ SL /@ DeleteDuplicatesBy[Llst, SL[#] &]

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