Find the eigenvalue of a stochastic matrix by only using linear operator
Solution 1:
The image of $A-I$ consists of vectors whose sum of coordinates is $0$. Indeed, if $v$ is any vector, then the first coordinate of $(A-I)v$ is $\langle r_1(A), v\rangle -v_1$ (dot product of the first row of $A$ and $v$ minus the first coordinate of $v$, the second coordinate is $\langle r_2(A), v\rangle -v_2$,..., the $n$-th coordinate is $\langle r_n(A), v\rangle -v_n$. Then the sum of coordinates is $\langle \sum r_i(A), v\rangle -\sum v_i=0$.
Thus $A-I$ is not surjective, as you wanted.