I have read that, for
$$I - \alpha P$$
where $I$ is the $n\times n$ identity matrix, $\alpha \in (0,1]$, and $P$ is the transition probability matrix with dimensions $n \times n$, $I - \alpha P$ is an invertible matrix because the eigenvalues of any transition probability matrix lie within the unit circle of the complex plane. "Thus no eigenvalue of $\alpha P$ can be equal to 1, which is a necessary and sufficient condition for $I - \alpha P$ to be invertible."

Why is that?


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