Is the matrix in the post invertible?

Question

Let me describe the matrix I am intersted in.

The square matrix $P\in R^{n\times n}$ with non-negative entries and satisfying:

$\forall 1\le i\le n, \sum_{j=1}^n P_{ij}<1$

I encounterd this matrix in studying the topic related to markov chains.

Is there any dedicated term for calling such matrix $P$?

The main question is:

"Is $P-I$ invertible?"

I have tried to show $1$ is not an eigen value of $P$ to prove its invertibility but still in trouble.

Any comments would be appreciated.

Answer 1

The Matrix is invertible. Suppose $Pv=v$. Take the maximum absolute entry of v, $v_k$. Then $$|v_k|=\left|\sum_{j=1}^{n}P_{kj}v_j\right|\leq |v_k|\sum_{j=1}^n|P_{kj}|<|v_k|$$ which means $v_k=0$, so $v=0$, thus 1 is no Eigenvalue of P.
(The first inequality holds by replacing every $v_j$ by the bigger $v_k$, as $v_k$ is the maximum).