Let $A = (a_{ij})_{i,j=1,\dots,n}$ be a matrix such that
$a_{ij} \ge 0$ for all $i,j = 1, \dots, n,$ and
$A$ is positive definite.
Let $I$ be the identity matrix, and $e$ the vector containing only ones.
Suppose that the solution $x = (x_i)_{i=1,\dots,n}$ to the system of linear equations
$$
(A+I)x=e
$$
is non-negative, i.e. $x_i \ge 0$ for $i=1,\dots,n$.
Define $f(t) = \frac{e^t + 1}{e^t - 1}.$
Is the solution $x$ to the system of linear equations
$$
(A+f(t)I)x=e
$$
non-negative for every $t>0$?