TITLE: Find $\det A$ and $\operatorname{Tr} A$ if $\det(A-\sqrt[n]{3}I_n)=0$
QUESTION [5 upvotes]: $A \in M_{n}(\mathbb{Q})$ and I have to find $\det A$ and $\operatorname{Tr} A$ if $\det(A+\sqrt[n]{3}I_n)=0$. I observed that $\sqrt[n]{3}$ is an eigenvalue of $A$,but I don't know how to continue.
EDIT : My bad,the matrix has rational entries.

REPLY [2 votes]: $\det (A+\lambda I_n)$ is a monic polynomial of degree $n$ in $\lambda$. Note that if $n$-th root of $3$ is a root of a degree $n$-polynomial, the polynomial must be a multiple of $\lambda^n-3$, hence $\det$ and Tr must be $(-1)^n\cdot -3$ and $0$ respectively (by using the fact that Tr is the coefficient of $\lambda^{n-1}$ in the characteristic polynomial and $\det$ is the constant term times $(-1)^n$ in the characteristic polynomial).