  It is shown that for positive real numbers $
0<\lambda_{1}<\dots<\lambda_{n}$, $\left[\frac{1}{\beta({\lambda_i},
{\lambda_j})}\right]$, where $ \beta(\cdot,\cdot)$ denotes the beta function,
is infinitely divisible and totally positive. For $
\left[\frac{1}{\beta({i},{j})}\right]$, the Cholesky decomposition and
successive elementary bidiagonal decomposition are computed. Let $\mathfrak
w(n)$ be the $n$th Bell number. It is proved that $\left[\mathfrak
w(i+j)\right]$ is a totally positive matrix but is infinitely divisible only
upto order $4$. It is also shown that the symmetrized Stirling matrices are
totally positive.
