Characteristic Equation is $ A -\mu I = 0$
$ \begin{vmatrix} 1-\mu & 1\\1 &-1-\mu\end{vmatrix} = 0$
$\implies (1-\mu)(-1-\mu)-1=0$
$\implies -1-\mu+\mu+\mu^{2}-1=0$
$\implies \mu^{2}-2=0$
$\implies \mu=+\sqrt 2 $ and $-\sqrt 2$
so according to properties of Eigen values,
eigen values of $A^{19}$=(eigen value of A)$^{19}$
=$(\sqrt 2)^{19}$ and $(-\sqrt 2)^{19}$
Hence Ans is option (D).