Integer solution to linear system

Question

Suppose $A \in \mathbb{R}^{n \times m}$ is a fat matrix ($n<m$) with full rank, and that all elements of $A$ are $1,-1$ or $0$. Given an integer vector $b \in \mathbb{Z}^n$, does there exist an integer vector $x \in \mathbb{Z}^m$ such that $Ax = b$?

Answer 1

How about $$A=\pmatrix{1&1&1\\1&1&-1}$$ and $$b=\pmatrix{1\\0}.$$ Any integer vector solution to $Ax=b$?