I have a row-stochastic matrix $G\in \mathbb{R}^{n\times m}$, such that $n<m$ and $Rank(G)<n.$ Define $J\in \mathbb{R}^{n\times m}$ as follows.
$$ J_{i,j} = \begin{cases} 1 & i=j,i\in [n]\\ 0 & \text{otherwise} \end{cases} $$
Is it true that the rank of $(1-\epsilon)\times G+\epsilon\times J$ for $\epsilon>0$ is $n$?
No. Here is a random counterexample generated by computer: $$ \epsilon=\frac15\ \text{ and }\ G=\frac{1}{12}\pmatrix{4&4&1&3\\ 1&1&7&3\\ 4&4&1&3}. $$ We have \begin{aligned} \pmatrix{1&1&-2}\left[(1-\epsilon)G+\epsilon J\right] &=\frac{1}{5}\pmatrix{1&1&-2}(4G+J)\\ &=\frac{1}{15}\pmatrix{1&1&-2}(12G+3J)\\ &=\frac{1}{15}\pmatrix{1&1&-2}\pmatrix{7&4&1&3\\ 1&4&7&3\\ 4&4&4&3}\\ &=0. \end{aligned}