1. ## Proving

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

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

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

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

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

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

Can anyone help me with part (b)?
I know that lambda must satisfy $Ax=\lambda x$

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

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

2. Originally Posted by Roam
(a) Show that if P is a non-singular nxn matrix then $\lambda$ is an eigenvalue of A if and only if, $\lambda$ is an eigenvalue of $PAP^{-1}$.
if $Ax=\lambda x,$ put $y=Px.$ then $PAP^{-1}y=\lambda y.$ conversely if $PAP^{-1}x=\lambda x,$ then put $P^{-1}x=y$ and see that $Ay=\lambda y.$ in each case it's important to note that $x \neq 0$ iff $y \neq 0.$

(b) Show that if P is a non-singular nxn matrix such that $A^3=o_{n}$. Show that the only possible eigenvalue for A is $\lambda = 0$. Here $0_{n}$ is the nxn zero matrix.
if $Ax=\lambda x,$ then $A^2x=A\lambda x=\lambda Ax=\lambda^2 x.$ repeat this to get $0=A^3x=\lambda^3x,$ and thus $\lambda = 0,$ because $x \neq 0.$