Special linear group of degree three or higher is not ambivalent

Statement

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

Proof

Case $n=3$

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

Explicitly, we could choose:

$A={\begin{pmatrix}0&0&1\\1&0&0\\0&1&1\\\end{pmatrix}},A^{-1}={\begin{pmatrix}0&1&0\\-1&0&1\\1&0&0\\\end{pmatrix}}$

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:

$B={\begin{pmatrix}A&0\\0&I_{n-3}\\\end{pmatrix}},B^{-1}={\begin{pmatrix}A^{-1}&0\\0&I_{n-3}\\\end{pmatrix}}$