Which of the following statements are true? (Select all that apply)

(i) A matrix is singular if and only if 0 is an eigenvalue

(ii) If $\det(A) = 0$, then $A$ has eigenvalue 0

(iii) If $A$ has eigenvalue 0, then $\det(A) = 1$

(iv) The product of all eigenvalues equals $\det(A)$