  Given an arithmetical function $f$, by $f(a, b)$ and $f[a, b]$ we denote the
function $f$ evaluated at the greatest common divisor $(a, b)$ of positive
integers $a$ and $b$ and evaluated at the least common multiple $[a, b]$
respectively. A positive semi-definite matrix $A=(a_{ij})$ with $a_{ij}\ge 0$
for all $i$ and $j$ is called infinitely divisible if the fractional Hadamard
power $A^{\circ r}=(a_{ij}^r)$ is positive semi-definite for every nonnegative
real number $r$. Let $S=\{x_1, ..., x_n\}$ be a set of $n$ distinct positive
integers. In this paper, we show that if $f$ is a multiplicative function such
that $(f*\mu)(d)\ge 0$ whenever $d|x$ for any $x\in S$, then the $n\times n$
matrices $(f(x_i, x_j))$, $(\frac{1}{f[x_i, x_j]})$ and $(\frac{f(x_i,
x_j)}{f[x_i, x_j]})$ are infinitely divisible. Finally we extend these results
to the Dirichlet convolution case which produces infinitely many examples of
infinitely divisible matrices. Our results extend the results obtained
previously by Bourque, Ligh, Bhatia, Hong, Lee, Lindqvist and Seip.
