If the augmented matrix has no trivial solution then we cannot find the value of c in the real field. As it is coming c^2+4=0 i.e. c=+-(2i).

If we prepare the augmented matrix and reduce by elementary operations we get 4-c^2=0 or c=+2 and c=-2.

For above values of c, the system has no trivial solutions. As we are getting a 2by2 matrix for three variables.