If A1 and A2 are irreducible row stochastic matrices, can we prove that (I-A1)(A2-I) is stable?

Question

$A_1\in \mathcal{R}^{n\times n}$ and $A_2\in \mathcal{R}^{n\times n}$ are row stochastic (row-sum-1) but not necessarily symmetric, and $I\in \mathcal{R}^{n\times n}$ is the identity matrix. Can we prove that $(I-A_1)(A_2-I)$ 1)has a single eigenvalue 0 and 2) the rest eigenvalues all have negative real parts.

I generate random matrix $A_1$ and $A_2$ and it seems that this property always holds, but I just cannot prove it. If the property is wrong, can you give a counterexample?

Thanks.