$\det(A-\lambda I)=0$
$\implies \begin{vmatrix} a-\lambda &1 &0 \\ 1 & a-\lambda &1 \\ 0& 1 & a-\lambda \end{vmatrix} = 0$
$\implies(a-\lambda)*[(a-\lambda)*(a-\lambda)-1*1] -1*[1*(a-\lambda)-0*1]+0*[1*1 - 0*(a-\lambda)] = 0$
$\implies(a-\lambda)^{3} - 2(a-\lambda)) = 0$
$\implies(a-\lambda)((a-\lambda)^2- 2) = 0$
$\implies(a-\lambda)((a-\lambda)^{2}- (\sqrt{2})^{2}) = 0$
$\implies(a-\lambda)(a- \lambda + \sqrt{2})(a-\lambda-\sqrt{2}) = 0$
Eigen values , $\lambda = a , a+\sqrt{2} , a - \sqrt{2}.$
Correct Answer: $A$