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 3×3 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.
Best Answer
One way to find the determinant is to bring the matrix in to row echelon form by row operations (which do not alter the determinant) and multiply the diagonal entries. If the determinant is $0$, one diagonal entry must be zero, and you can work your way back up the triangular matrix to obtain a solution with at least one "degree of freedom", i.e. , there is a nontrivial solution.