Given a Laplacian matrix $A\in\mathbb{R}^{n\times n}$ (symmetric, positive semi-definite matrix with positive diagonal elements and non-positive off-diagonal), and its Moore-Penrose pseudoinverse $A^+$, why is $A^+Ab=b,$ for some vector $b\in\mathbb{R}^n$ whose elements sum to zero. Note that the null-space of $A$ is spanned by vector of all ones, $1_n=(1,1,...1)\in\mathbb{R}^n$.
I find that $A^+=(A+1_n1_n^T)^{-1}-n^{-2}1_n1_n^T$, but the second term will vanish in multiplication with $A$.