TITLE: Is $A + A^{-1}$ always invertible?
QUESTION [18 upvotes]: Let $A$ be an invertible matrix. Then is $A + A^{-1}$ invertible for any $A$? 
I have a hunch that it's false, but can't really find a way to prove it. If you give a counterexample, could you please explain how you arrived at the counterexample? Thanks.
This isn't HW, and I don't really have any work to show.

REPLY [14 votes]: As the other answers show, the answer is no. However, if $A$ is symmetric it is true (Hermitian in the complex case): Let $A$ have eigenvalues  $a_i$, and note that $A+A^{-1}$ has eigenvalues $a_i + a_i^{-1}$. These are zero if and only if $a_i^2 = -1$. But this is not possible since $a_i$ is real. Hence $A + A^{-1}$ is invertible.