Let $A$ be an $n\times n, (n\ge2)$ matrix with char poly $x^{n-2}(x^2-1)$ Then which of The following is true?
$A^n=A^{n-2}$,
$r(A)=2$,
$r(A)$ is atleast $2$,
there exist non zero vector $x,y$ such that $A(x+y)=x-y$,
Well, I can only see that 1 is true from the caley Hamilton Theorem, would you help other three are correct or not?