Suppose that a1,a2,…,an are n distinct real numbers; is the following statement true?
There is a permutation of a1,a2,…,an, namely b1,b2,…,bn, such that the determinant of the following matrix is nonzero:
[b1b2⋯bnb2b3⋯b1⋮⋮⋱⋮bnb1⋯bn−1]
(Such a matrix is called a circulant matrix.)
Answer
This statement is not true, without supplementary conditions on the ai's. Indeed,
suppose the ∑nk=1ak=0, whatever your permutation is the vector [1,1,…,1]T is in the kernel of the circulant matrix of the bi's, and consequently, its determinant is 0.
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