sign the elements of a ugly matrix

Question

I have a matrix defined as $A=(I-aT)^{-1}F$, where I is identity matrix, a is a positive constant smaller than 1, T is a stochastic (transition matrix) and F is a matrix with positive diagonal elements and negative elements elsewhere. All matrix are $n*n$. I want to sign the elements in A. I got stuck somewhere by the inverse matrix. I am wondering are there some clear theorem related to show that? Or what else conditions do I need to show that? Thank.

Answer 1

$(I-aT)^{-1}=\sum_{k=0}^{+\infty}a^kT^k....$