While I was solving problem to determine if given set of vectors are linearly independent or not, the solution given was that the coefficient matrix A of the homogeneous linear system of those vectors
k1a1 + k2b1 + k3c1 = 0
k1a2 + k2b2 + k3c2 = 0
k1a3 + k2b3 + k3c3 = 0
was a 3x3 square matrix, and determinant(A) = 0
Thus it has a non trivial solution and hence the vectors are linearly dependent.
I am unaware about how if A is nxn square matrix and if determinant(A) = 0, then the system is having a non trivial solution.