If $A^k$ is the identity matrix for some $k \geq 2$, then $A^2$ is the identity matrix

Question

Let $A$ be symmetric matrix.

Show that if $A^k$ is the identity matrix for some $k \geq 2$, then $A^2$ is the identity matrix

Show that if $A^k$ is the zero matrix for some $k≥2$, then A is the zero matrix.

Could anyone give me a hint of how to start this? Should I start with idempotent, eigenvalues?

Answer 1

Hints for the first part: If $A$ is symmetric, it is orthogonally diagonalisable, so $A = V D V^{-1}$. How does $A^k$ look like? Now try to conclude that $A^k = I$, where $I$ is the identity, implies that $D^k = I$. What can you now say about the eigenvalues of $A$, which are the diagonal entries of $D$? Lastly, now that we know how $D$ and $D^2$ looks like, how does $A^2$ look like?

The second part is very similar.