Matrix solutions of the equation \(X^ n=A\) (Q1174834)

The matrix equation \(X^ n=A\) in \(M_ 2(\mathbb{Z})\) for given \(A\) and \(n\in\mathbb{N}\) is studied. If \(A\neq aI\) for integer \(a\), then there are only finitely many solutions. An explicit algorithm is given to determine them. If \(A=aI\), there can be infinitely many solutions. In particular, given a matrix \(A=\left[{a\atop c}{b\atop d}\right]\in M_ 2(\mathbb{Z})\), \(\det(A)\neq 0\), and an integer \(n>1\), the matrices \(P=\left({p\atop r}{q\atop s}\right)\in M_ 2(\mathbb{Z})\), such that \(P^ n=A\) are characterized. In the case that \(A=aI\) implying infinitely many associated solutions, a classification regarding similarity over \(\mathbb{Q}\) can easily be given. In fact, if there are infinitely many solutions, the roots are classified up to similarity over \(\mathbb{C}\) and the classification regarding similarity over \(\mathbb{Z}\) is then reduced to \(\mathbb{Z}\)-similarity classification of matrices in \(M_ 2(\mathbb{Z})\) having a fixed characteristic polynomial.