Matrices manipulation

Question

I am having difficulty with the following question

I have to determine if the following claim is true or not.

If it is true I have to proof it else I need to give an example

I believe it is not true but I do not find.

$A$ is a matrix (with scalar in real).

$A$ is symmetric.

if there exists $k>0$ so that $A^{k}=Id$ then $A^{2}=Id$

how can I proof if it is not true ?

in order to proof do I need to assume that $k=3$ and $A^{2}!=Id$ ?

although I still believe this is not true

Thanks in advance

Answer 1

Hints: a real symmetric matrix is diagonalizable (via an orthogonal matrix, but you don't even need that here) in $M_n(\mathbb{R})$ and the condition $A^k=Id$ implies that the eigenvalues of $A$, which are all real, belong to $\{\pm 1\}$. Finally, note that the result is true.