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?

Yes. Moreover, it must be an integer. 
Let $v$ be an eigenvector of A, then:
$Av=\lambda v$
for some lambda. Multiplying by A we get:
$A^2v=Iv=v=\lambda^2v$
Which means that any eigenvalues $\lambda_n$ satisfy $\lambda_n^2=1$ and thus $\lambda_n=\pm1$. Recall that the trace of a matrix is the sum of the eigenvalues of the matrix, which directly implies the result. 
