The row- and column-sums of a nonengative matrix with spectral radius less than $1$

Question

Is it true that if a matrix has nonnegative elements and spectral radius less than $1$, than the sum of its elements on each row (and column) is less than $1$?

Edit: What if the matrix has positive elements?

Answer 1

$A=\begin{bmatrix}0 & 1 \\ 0 & 0 \end{bmatrix}$. The spectral radius is $0$, but a row sum is $1$.

A counterexample with all entries strictly positive follows immediately from continuity of eigenvalues.

For an explicit example, try $A=\begin{bmatrix}\frac{1}{10} & 1 \\ \frac{1}{10} & \frac{1}{10} \end{bmatrix}$. It is straightforward to check that $\rho(A) = \frac{\sqrt{10}+1}{10}<1$.

For a symmetric example, take $A=\begin{bmatrix}\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{20} \end{bmatrix}$. This has $\rho(A) = \frac{\sqrt{481}+11}{40}<1$.