Let $A$ be $6 \times 6$ matrix with $A^5 = I$, Find the rank($A^2$).
The only thing I can think about this question is $A^5 = I$ implies $x^5 - 1 =0$ is annihilating polynomial. Can I conclude or continue solving the problem with that idea ?
How can we solve these type of problems. What are the results I have to know. Is there a general idea for solving them ??
We have
$$A^2A^3=A^3A^2=A^5=I.$$
Thus, $A^2$ is inverible. Conclusion ?