Show that the matrix $\big(\frac{1}{a_i + a_j}\big)_{n\times n}$ is positive semidefinite

Question

Given positive real numbers $a_1,\dots ,a_n$, how can I prove that the symmetric matrix composed of the entries $\frac{1}{a_i + a_j}$ is positive semi-definite

*this is not a hw question but a self study question from the matrix theory book by xinghzi zhan.