When solving a problem about linearly independent/dependent, I don't know what the answer means.

Question: Determine whether the collection of vectors is linearly independent in $R^3$:
$(1,2,4)^T$ , $(2,1,3)^T$ , $(4,-1,1)^T$

Answer:
If $$c\_1(1,2,4)^T+c\_2(2,1,3)^T+c\_3(4,-1,1)^T=(0,0,0)^T$$

Then,
$$\begin{array}{rcl}
c\_1+2c\_2+4c\_3=0\\
2c\_1+c\_2-c\_3=0\\
4c\_1+3c\_2+c\_3=0\\
\end{array}$$
The coefficient matrix of the system is singular and hence the system has nontrivial solutions. Therefore the vectors are linearly dependent.

I know the method to determine whether the matrix is linearly independent or not by computing $c\_1v\_1+c\_2v\_2+...c\_nv\_n=0$, but here I don't know why the system is singular, so how can I tell whether it's singular or not? Are there any ways to know?