The characteristic equation can be rewritten as :
λ3 + 2 λ2 + 2 λ + 1 = 0
==> λ ( λ2 + 2 λ + 1) + 1(λ + 1) = 0
==> λ (λ + 1)2 + 1(λ + 1) = 0
==> (λ + 1) (λ2 + λ + 1) = 0
Solving which we get λ = -1 , ω , ω2 where ω , ω2 are cube roots of unity ..
As we know :
Modulus of each of cube roots of unity = | ω | = | ω2 | = 1
Also we know ,
Eigen values of matrix satisfies the corresponding characteristic equation and if all eigen values have modulus value = 1 , then the matrix is said to be orthogonal.
which is the case here..
Hence C) is the correct answer..