Trace of a matrix $A$ with $A^2=I$ Let $A$ be a complex-value square matrix with $A^2=I$ identity.
Then is the trace of $A$ a real value?

Another way.
$$A^2-I=0,$$
Since minimal polynomial divide this polynomial (why) then it must be either
$(x-1)(x+1)$ or $(x-1)$ or $(x+1)$.
hence all the eigenvalues are reals (even in set $\{1,-1\}$) (why) and hence the result follows.
