  In this paper, we study the eigenvalues of the GCD matrix $(S_n)$ and the LCM
matrix $[S_n]$ defined on $S_n=\{1,2,\ldots,n\}$. We present upper and lower
bounds for the smallest and the largest eigenvalues of $(S_n)$ and $[S_n]$ in
terms of particular arithmetical functions.
