General linear group of degree three or higher is not ambivalent

Statement

Suppose $R$ is a commutative unital ring. Then, the general linear group of degree three $GL(3,R)$ is not an ambivalent group. In fact, for $n\ge 3$, the group $GL(n,R)$ is not an ambivalent group.

Proof

Case $n=3$

Given a matrix $A$ with characteristic polynomial $x^3+k{x}^2+lx+m$, with $m$ invertible, the characteristic polynomial of $A^{-1}$ is $x^3+(l/m)x^2+(k/m)x+(1/m)$. We need to choose values of $k,l,m$ such that these characteristic polynomials are distinct. Consider the case $k=m=-1,l=0$. Thus, $A$ has characteristic polynomial $x^3-x^2-1$ and $A^{-1}$ has characteristic polynomial $x^3+x-1$.

Explicitly, we could choose:

$A=\left(\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 1\\ \end{array}\right),A^{-1}=\left(\begin{array}{ccc}0& 1& 0\\ -1& 0& 1\\ 1& 0& 0\\ \end{array}\right)$

These have different traces (1 and 0). Note that the example works even over field:F2 and rings of characteristic two.

Case of higher $n$

We can pad the example given above with the identity matrix, i.e., using $A$ and $A^{-1}$ as above, set:

$B=\left(\begin{array}{cc}A& 0\\ 0& I_{n-3}\\ \end{array}\right),B^{-1}=\left(\begin{array}{cc}{A}^{-1}& 0\\ 0& I_{n-3}\\ \end{array}\right)$