Trace of $A$ if $A =A^{-1}$ Suppose $I\neq A\neq -I$, where $I$ is the identity matrix and $A$ is a real $2\times2$ matrix. If $A=A^{-1}$ then what is the trace of $A$? I was thinking of writing $A$ as $\{a,b;c; d\}$ then using $A=A^{-1}$ to equate the positions but the equations I get suggest there is an easier way.

Generalizing if $A^2=\operatorname{Id}$ we have that $\operatorname{tr}(A)= \dim(V_1)-\dim(V_{-1})$.
That's because $A$ is diagonalizable and trace is invariant for conjugacy. 
