  In this paper we study the structure and give bounds for the eigenvalues of
the $n\times n$ matrix, which $ij$ entry is $(i,j)^\alpha[i,j]^\beta$, where
$\alpha,\beta\in\Rset$, $(i,j)$ is the greatest common divisor of $i$ and $j$
and $[i,j]$ is the least common multiple of $i$ and $j$. Currently only
$O$-estimates for the greatest eigenvalue of this matrix can be found in the
literature, and the asymptotic behaviour of the greatest and smallest
eigenvalue is known in case when $\alpha=\beta$.
