Proof spectral radius less than $1$

Question

Given that matrix $A$ is set as:
$A=(I-P)(I-QP)^{-1}$
where matrix $QP$ is non-negative reducible hollow matrix, and $\rho(QP)<1$. Matrix $P$ is a diagonal matrix and all entries in $P$ are in interval $[0,1]$
I am trying to proof that the spectral radius $\rho(A)<1$. I used Neumann Series to rewrite the matrix $(I-QP)^{-1}$ as
$\sum_{k=0}^{\infty}(QP)^k$, then I am stuck here. How could I proof that the spectral radius of matrix $A$ is less than $1$? Any help will be appreciated!