It is well known that for $A \in \mathbb{K}^{n \times m}$ and $b \in \mathbb{K}^n$ the system $$ Ax = b $$ is solvable if and only if $\operatorname{rank}(A) = \operatorname{rank}(A \mid b)$.
I suppose, in practice one would just use Gauß-Algorithm. However, is there any criterion for $\operatorname{rank}(A) = \operatorname{rank}(A \mid b)$?