I have to prove that if $V$ is a finite-dimensional vector space over a field of characteristic not 2, and $T$ is an endomorphism such that $\det(I+T) \neq 0$ then $T \mapsto (I-T)(I+T)^{-1}$ is an involution on the space of endomorphisms such that $\det(I+T) \neq 0$.
This is part of an exercise in Gadea and Masqué workbook. It seem trivial as they don't bother detailing the answer, but I can't find it.
Thanks,
JD