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Find determinant of matrix obtained by two permuations

Source: 2018 South Korea USCM P4

August 14, 2020
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Problem Statement

n≥2n\geq 2 is a given integer. For two permuations (α1,⋯ ,αn)(\alpha_1,\cdots,\alpha_n) and (β1,⋯ ,βn)(\beta_1,\cdots,\beta_n) of 1,⋯ ,n1,\cdots,n, consider n×nn\times n matrix A=(aij)1≤i,j≤nA= \left(a_{ij} \right)_{1\leq i,j\leq n} defined by aij=(1+αiβj)n−1a_{ij} = (1+\alpha_i \beta_j )^{n-1}. Find every possible value of det⁡(A)\det(A).
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