Let $A = \left[{\begin{matrix}2 & 4 \cr 5 & 10 \cr \end{matrix}}\right]$.

We want to determine if the equation $A{\bf x}={\bf 0}$ has nontrivial solutions.
To do that we row reduce $A$.

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

A. The equation has no nontrivial solutions.
B. The equation has nontrivial solutions.
C. We cannot tell if the equation has nontrivial solutions or not.