When is an invertible $n \times n$ matrix (over an arbitrary ring) similar to its inverse?

Question

I want to find necessary and sufficient conditions for an invertible $n\times n$ matrix (over an arbitrary ring) similar to its inverse. Two $n\times n$ matrices $A$ and $B$ are called similar if there exists an invertible $n\times n$ matrix $P$ such that $B=P^{-1}AP.$ I tried to find it with invertible real 2 × 2 matrices. But I did not get any results. I don't know if anyone has any results on this problem.

Answer 1

Note that if $p(x)$ is the characteristic polynomial of $A$, then the characteristic polynomial of $A^{-1}$ is $x^n p(\frac 1x)/\det A$; therefore a necessary condition is that $x^n p(\frac 1x) = p(x) \det A$.

Note also that if $A$ is diagonalizable, then the condition that the reciprocal of every eigenvalue of $A$ is also an eigenvalue of $A$ (respecting multiplicity) is both sufficient and necessary. One can dispense with the diagonalizability assumption if one is willing to consider the blocks in the Jordan canonical form of $A$.

Answer 2

There is a general result of Wonenburger that a matrix in $M_n(F)$ ($\operatorname{char}(F) \ne 2$) that is similar to its inverse is a product of two involutions. (the converse: $ (u v)^{-1} = v u = v (u v) v^{-1}$, is valid in any group).