Be it real matrix or complex matrix, the identity matrix remains the same. So we will have in $I_n,$ $1's$ only in the principal diagonal elements and the rest of the elements of the matrix will be $0.$ So the characteristic equation will be :
$\mid A - λ I \mid = 0$
$\implies \underbrace{(1 - λ) . (1 - λ) \ldots (1 - λ)}_{n \text{ times}} = 0 $
$\implies (1 - λ)^n = 0$
$\implies λ = 1$ as the only solution the reason being complex roots of unity only holds if it were $λ^n - 1 = 0$ which is not the case here.
Hence (E) is the correct answer.