When can I use the determinant to determine whether vectors are linearly independent?
You can use the determinant to check if vectors are linearly independant in the following cases:
If you are not in the cases mentioned above, I would recommend you go back to the definition. Does the equation
$$\lambda_1X_1+\lambda_2X_2+\cdots\lambda_nX_n=0$$ have a unique solution?
if yes, they are linearly independent.
If no, they are linearly dependent.
Remember that the coefficients \(\lambda_i\) are constant, They are not variable functions.