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?

$$A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\implies A^2=\begin{pmatrix}a^2+bc&b(a+d)\\c(a+d)&d^2+bc\end{pmatrix}=\begin{pmatrix}1&0\\0&1\end{pmatrix}\implies$$
$$\begin{align}a^2+bc=1=d^2+bc\implies a=\pm d\end{align}$$
But we also have that
$$b(a+d)=0=c(a+d)\implies \begin{cases}a=-d\\or\\b=c=0\end{cases}$$
In the first case
$$a=-d\implies \text{Tr.}\,A=0$$
In the second case:
$$b=c=0\implies a=\pm d=\pm 1\implies \;\text{Tr.}\,A=0\;\;or\;\;\text{Tr.}\,A=\pm2$$
