I've been studying a symmetric matrix with the following form:
$$ \begin{bmatrix} a_1 & a_2 & a_3 & a_4 & \cdots & a_n \\ a_2 & a_2 & a_3 & a_4 & \cdots & a_n \\ a_3 & a_3 & a_3 & a_4 & \cdots & a_n \\ a_4 & a_4 & a_4 & a_4 & \cdots & a_n \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ a_n & a_n & a_n & a_n & \cdots & a_n \end{bmatrix} $$
where $a_i > 0$ for $i = 1, \dots, n$.
I was curious if a matrix with this form has been described previously and, if so, what properties it has, particularly its inverse.