Why is the vector $(I-A)^{-1}x$ positive?

Question

Why is the vector $(I-A)^{-1}x$ positive?

I was reading a text where they make the conclusion that the product $(I-A)^{-1}x$ is a vector with positive entries.

Notes:

1. $I$ is the identity $n\times n$ matrix.
2. $A$ is a $n\times n$ matrix with zeros on the diagonal and non-negative entries such that $\|A\|_{\infty}<1$, thus $I-A$ is invertible.
3. $x$ is a vector with positive entries.

Answer 1

By definition, we have \begin{align} (I-A)^{-1}=I+A+A^2+\cdots+A^n+\cdots. \end{align} Next, we see that $A^kx$ has positive entries since $A^k$ is a matrix with nonnegative entries and $x$ is a positive vector. Hence $(I-A)^{-1}x$ is a positive vector.