Which of the following statements is correct?

Here are some matrices to think about.

$A = pmatrix {1\&1\&&-1}$

$A = pmatrix {cos frac {2pi}{3}& sin frac {2pi}{3}\-sin frac {2pi}{3}&cos frac {2pi}{3}}$

$A = pmatrix {1&&\&cos frac {2pi}{3}& sin frac {2pi}{3}\&-sin frac {2pi}{3}&cos frac {2pi}{3}}$

In all of the scenarios

$A^n = I$ says something about the eigenvalues of $A$

Then there is a further clause that restricts what those eigenvalues may be.

And finally, do all matrices that meet the previous 2 constraints meet the 3rd constraint.

Update.

$A = pmatrix {-frac 12& frac {sqrt 3}{2}\-frac {sqrt 3}{2}&-frac 12}$

$A^2 = pmatrix {-frac 12& -frac {sqrt 3}{2}\frac {sqrt 3}{2}&-frac 12}$

$A^2+A+I = 0$