RO-NMO 2005 Problem 111

Let $n\geq 2$ a fixed integer. We shall call a $n\times n$ matrix $A$ with rational elements a radical matrix if there exist an infinity of positive integers $k$ , such that the equation $X^k=A$ has solutions in the set of $n\times n$ matrices with rational elements.

a) Prove that if $A$ is a radical matrix then $\det A \in \{-1,0,1\}$ and there exists an infinity of radical matrices with determinant 1; b) Prove that there exist an infinity of matrices that are not radical and have determinant 0, and also an infinity of matrices that are not radical and have determinant 1.

After an idea of Harazi