An eigenvector of a linear operator is a nonzero vectors s.t. the operator multiplies it by a scalar. It's existence for a complex finite dimensional operator c

An eigenvector of a linear operator is a nonzero vectors s.t. the operator multiplies it by a scalar. It's existence for a complex finite dimensional operator can be proven without the determinant. The minimal and the characteristic polynomials can be defined and described without the determinant. Where do you think you need the determinant?