How to prove that $\,I-\gamma\, T\,\;\big(\,T\,$ is stochastic matrix, $\,0 \le \gamma \lt 1\,\big)\,$ is invertible

Question

In case $\,T\,$ is right stochastic matrix (sum of each row is $1$) and $\,0 \le \gamma \lt 1$,

Is there any way to prove that $\,I-\gamma \,T\,$ is invertible?

Answer 1

Two helpful facts:

• Any eigenvalue of $I-\gamma T$ is of the form $1-\gamma \lambda$ where $\lambda$ is an eigenvalue of $T$.
• All eigenvalues of $T$ are in $[-1,1]$.

You should prove these two facts and then finish the proof in one more step.

(Alternately, the way you prove the second statement could be used to directly prove the original statement, by investigating $I-\gamma T$ instead of $T$ itself.)