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:

- n constant vectors of dimension n
- n vector functions of dimension n if they are solution to the same homogeneous system X'=AX. In this case the determinant is called the Wronskian.

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.