Let $A = \left[{\begin{matrix}2 & 4 & 2 \cr 3 & 9 & 18 \cr 4 & 10 & 16 \cr \end{matrix}}\right]$.

We want to determine if the equation $A{\bf x}={\bf b}$ has a solution for all ${\bf b}$.
To do that we row reduce $A$.

To do this we add times the first row to the second.
We then add times the first row to the third.
We then add times the new second row to the new third row.
We conclude that

A. The equation has a solution for all ${\bf b}$.
B. The equation does not have a solution for all ${\bf b}$.
C. We cannot tell if the equation has a solution for all ${\bf b}$ or not.