Invertibility of this matrix

Question

Let $P_1$ and $P_2$ be two stochastic matrices. Prove that $I - P_1 + P_2$ is invertible. I know that the eigenvalues of $P_1$ and $P_2$ are at most 1 in magnitude. How can I handle the difference $P_2 - P_1$ to show that the eigenvalues of $I - P_1 + P_2$ can never be zero, and thus $I - P_1 + P_2$ is invertible?

Answer 1

Claim isn't true.

$$\left[ \begin{array}\ 1&0&0\\ 0&1&0\\ 0&0&1 \end{array} \right]-\left[ \begin{array}\ 1&0&0\\ 0&1&0\\ 0&0&1 \end{array} \right] +\left[ \begin{array}\ 0&0&0\\ 0&0&0\\ 1&1&1 \end{array} \right]=$$
$$ \left[ \begin{array}\ 0&0&0\\ 0&0&0\\ 1&1&1 \end{array} \right] $$

which is not invertible.