Determine values of the constant k for which vectors are linearly independent?

Problem:

Determine all values of the constant k for which the given set of vectors is linearly independent in R4.

{(1,0,1,k),(-1,0,k,1),(2,0,1,3)}

Attempt:

So far, I have the system of equations:

x-y-2z=0

x+ky+z=0

kx+y+3z=0

Now, the determinant of the coefficient matrix yields 2(k+1)(2-k).

Here is my question: How does the determinant of the coefficient matrix allow me to deduce that "the system has only the trivial solution, and the vectors are linearly dependent if and only if k=-1 or 2"?

Thanks for the help! :)