Vieta Matrix and its Determinant

Abstract

In this paper, we define a matrix which we call Vieta matrix and calculate its determinant: $$ \left( \begin{array}{cccc} 1&1&\cdots&1\\ a_{2}+a_{3}+\cdots+a_{n}&a_{1}+a_{3}+\cdots+a_{n}&\cdots&a_{1}+a_{2}+\cdots+a_{n-1}\\ a_{2}a_{3}+a_{2}a_{4}+\cdots+a_{n-1}a_{n}&a_{1}a_{3}+a_{1}a_{4}+\cdots+a_{n-1}a_{n}&\cdots&a_{1}a_{2}+a_{1}a_{3}+\cdots+a_{n-2}a_{n-1}\\ \vdots&\vdots&\ddots&\vdots\\ a_{2}a_{3}\dots a_{n-1}a_{n}&a_{1}a_{3}\dots a_{n-1}a_{n}&\cdots&a_{1}a_{2}\dots a_{n-2}a_{n-1} \end{array} \right). $$