(a) Show that if P is a non-singular nxn matrix then $\displaystyle \lambda$ is an eigenvalue of A if and only if, $\displaystyle \lambda$ is an eigenvalue of $\displaystyle PAP^{-1}$.

(b) Show that if P is a non-singular nxn matrix such that $\displaystyle A^3=o_{n}$. Show that the only possible eigenvalue for A is $\displaystyle \lambda = 0$. Here $\displaystyle 0_{n}$ is the nxn zero matrix.

For part (a) I think I need to manipulate the expression, here's what I think:

$\displaystyle

PAP^{-1} (x) = \lambda x

$

$\displaystyle A(PxP^{-1}) = \lambda x$

$\displaystyle A(x)PP^{-1} = A(x) I_{n} = A(x) = \lambda x$

Can anyone help me with part (b)?

I know that lambda must satisfy $\displaystyle Ax=\lambda x$

we have: $\displaystyle A.A.A (x) = \lambda x$

I don't know what to do with this & I'm I'm really stuck here...