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

No. Take $A=I_m$ and $\mathbf{b}=0$. The homogeneous linear system then has only the solution $\mathbf{x}=0$. Homogeneity ensures existence, but infinitely many solutions would additionally require a nonzero kernel.