Find a real symmetric matrix $A$

Question

Is there a real symmetric matrix $A$ satisfying the following two conditions?

1. $A$ is not orthogonal.
2. There is a positive integer $m >1$, such that
   $$A^m=I.$$

Answer 1

We have that

$$A^m=I \implies A^2=I \iff A^TA=I$$

Refer to the related

• Real Symmetric Matrices