Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics I / 2 / b / i

No. Here $A$ has more rows than columns, which does not force a nonzero kernel. For example, with $m=1$, $n=2$,the linear system has the unique solution $x=0$. With the same $A$ and $\mathbf{b}=(0,1{)}^T$ it has no solution at all.