Find Jordan Form if $A^2$=$E$

Question

Find Jordan Form of $A$ if $A^2$=$E$

$E$ - Identity matrix.

I was wondering how can I find characteristic polynomial from this conditions. Or probably there is a 'special' solving way, never met such tasks before.

Couldn't find a way or a tip on the internet too.

Answer 1

Note that the polynomial $$ p(t)=(t-1)(t+1) $$ annihilates $A$. It follows that the minimal polynomial $\mu_A(t)$ must divide $p(t)$. In particular, this implies that the only possible eigenvalues of $A$ are $\pm 1$. Moreover, the largest Jordan block for any eigenvalue has size one, since the minimal polynomial splits into linear factors. This implies that the Jordan form is $$ \DeclareMathOperator{diag}{diag} \diag(\pm1,\dotsc,\pm1) $$

Answer 2

Hint: $(A-E)(A+E)=A^2-E=0$. What can you say about the minimal polynomial of $A$?