Question:
If a square matrix $A$ satisfies $A^2=I$ and $\det A>0$, show that $A+I$ is non-singular.
I have tried to suppose a non-zero vector $x$ s.t. $Ax=x$ but fail to make a contradiction.
And I tried to find the inverse matrix of $A+I$ directly, suppose $(A+I)^{-1}=\alpha I +\beta A$, but it still doesn't work.
(Update: According to the two answers, this question itself is incorrect.)