$P$ is the transition probability matrix for a finite irreducible markov chain. $I$ is identitiy matrix. $A$ is the matrix whose entries are all $1$. Prove $I+A-P$ is invertible.
I don't have any clear idea. Apparently irreducibility is crucial but I just cannot find a way to use it.
Assume that $(I+A-P)x=0$ for some vector $x$ then $x+Ax=Px$. Note that $PA=A$ because $P$ is a transition matrix hence $P^kx+Ax=P^{k+1}x$ for every $k\geqslant0$, that is, $P^{k+1}x=x+(k+1)Ax$. Each coordinate of $P^{k+1}x$ stays bounded when $k\to\infty$ since every $P^{k+1}$ is a transition matrix, hence $Ax=0$ and $Px=x$. Since $P$ is irreducible, its eigenvalue $1$ is simple. Let $u$ denote the vector whose every coordinate is $1$, then $Pu=u$ hence $x$ is a multiple of $u$. But $Ax=0$ and $Au\ne0$ hence $x=0$.