I have a nonnegative real matrix $A$ for which all columns sum to 1 except for one which has all 0's. I need to show that $I - xA$ for some $x \in [0,1]$ is invertible.
I know that $I - B$ is invertible if the spectral radius $\rho(B) < 1$ exists and that the Perron-Frobenius theorem gives bounds for $\rho(B)$ for positive matrices: $\min_i \sum_j B_{ij} \le \rho(B) \le \max_j \sum_j B_{ij}$. However I don't know how to give bounds for non-negative matrices.
Hint: Let $v$ be an eigenvector of $A$ with corresponding eigenvalues $\lambda$, then we see that \begin{align} (I-xA)v = (1-x\lambda)v \end{align} which means $v$ is also an eigenvector with eigenvalues $1-x\lambda$. Choose $x$ to be small.