Matrix polynomial relation

Question

We have $A, B$ $n$ by $n$ matrices with real entries. Is $\det(AB+xBA)=\det(BA+xAB)$? Or is there a relation between them for $3$ by $3$ matrices?

Answer 1

One notable relation is that if $p(x) = \det(AB + xBA)$, then $$ x^n p(1/x) = x^n\det(AB + BA/x) = \det(ABx + BA) = \det(BA + xAB) $$ I see no reason that these two polynomials should coincide in general.

Answer 2

No, $\det(AB+xBA)$ is not equal to $\det(BA+xAB)$ in general. In fact, trying some random matrices I could not find even one example where the equality hold besides making $x=0$ or $x=1$.

Answer 3

if $A$ and $B$ commute, or if $x \in \{0,1\}$ then $\det(AB+xBA)=\det(BA+xAB)$