Positivity properties of some special matrices

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.