At the time of solving I just forgot this property & get two method, one method is working fine but other is not,but can't understand why the other method is not working.
1st Method:
By using Cayley-hamilton Theorem from the characteristic eqn I get $\lambda ^{2} - 2 = 0$ which is $A^{2} - 2 = 0$
$A^{2} = 2I \Rightarrow ({A^{2}})^9 = 2^{9}I$
i.e $A^{18}=512I$, Now post-multiplied by A on both side $A^{18}.A=512I.A$. which becomes $A^{19}=\begin{bmatrix} 512 &512 \\ 512&-512 \end{bmatrix}$
computing eigen values here we'll get $512\sqrt2$ & $-512\sqrt2$
This method is ok & it's working.
2nd method:
finding a pattern to compute A^19 which 1st compute A^2 by A*A , then A^3 by A^2 * A, then A^4 by A^3 * A
by this we'll get a pattern like,
$A^{n} =\begin{bmatrix} n &0 \\ 0 &n \end{bmatrix}$ when n is even
$A^{n} =\begin{bmatrix} n-1 &n-1 \\ n-1 &1-n \end{bmatrix}$ when n is odd
likewise $A^{19} =\begin{bmatrix} 18 &18 \\ 18 &-18 \end{bmatrix}$
but computing eigen values on $A^{19}$ will give $-18\sqrt2$ & $18\sqrt2$
don't know why this 2nd method is not working,Anyone please give a reason why this is not working.