Show that n $\sum_{i=1}^n c_i$ and $\prod_{i=1}^n c_i$ are rational numbers.

Question

Let $n$ be a positive integer and let $A ∈ M_{n×n}(\mathbb{Q})$. Let $c_1,...,c_n$ be the list of (not necessarily distinct) eigenvalues of $A$, considered as a matrix in $M_{n×n}(\mathbb{C})$. Show that $\sum_{i=1}^n c_i$ and $\prod_{i=1}^n c_i$ are rational numbers.

I'm not sure how to proceed, any solutions/hints are greatly appreciated. This result seems really interesting

Answer 1

Hint The sum and product of the eigenvalues are the trace respectively the determinant of the matrix.