is (I+P) invertible when row sum in P = 0

Question

I have a $n$x$n$ matrix P where the sum of each row = 0 (the individual entries are real but can be negative). Clearly P is not invertible. Can we show that I+P is invertible?

thanks

Answer 1

If $P = \left( \begin{array}{cc} -1 & 1 \\ 0 & 0 \\ \end{array} \right)$ then $P + I = \left( \begin{array}{cc} 0 & 1 \\ 0 & 1 \\ \end{array} \right)$ which is not invertible.

Answer 2

A matrix $P$ whose row-sum is zero will satisfy this statement if and only if $P$ does not have $-1$ as an eigenvalue.

If $P$ has a row-sum of zero, all that says is that $0$ is an eigenvalue of $P$ and that $\pmatrix{1&\cdots & 1}^T$ is an associated eigenvector.