In 1875, H.~J.~S. Smith proved the following remarkable result: the determinant of the $n\times n$ matrix whose entries are the GCDs $(i,j)$, for $1\leq i,j\leq n$, is equal to $\prod_{k=1}^n \varphi(k)$, where $\varphi$ is the Euler totient function [Proc. Lond. Math. Soc. {\bf 7} (1875/76), 208--212; MR1575630]. \par Many further extensions of this result have been given since then, considering general arithmetical functions and more general rings such as UFDs (see the references of this article). \par This article contains a generalization of related formulas for the $n\times n$ power GCD matrix over a UFD $R$ (i.e., the entries are the same power of the GCDs $(i,j)$ in $R$). Suitable definitions for this setting are introduced and recalled, such as the Möbius function, the Dirichlet convolution and the Jordan function.