There exists a $2 \times 2$ real matrix $A$ such that $A^5 = I$.
a) $A$ must be identity.
b) $A$ must be similar to an element of $SO(2)$ .
c) $A$ must be diagonalisable.
I have checked that minimal polynomial must divide $(x^5-1)=(x-1)(x^4+\dots+1)$ so it must be $x-1$ so $A=I$, and of course diagonalizable so a and c are correct options, but I am not sure about b.