It is known that for matrices $A$ and $B$, $AB$ and $BA$ are isospectral. However, I am not sure if this holds for a when these terms are in a sum, i.e. if $AB+C$ is isospectral to $BA+C$. My intuition tells me that I shouldn't expect these to have the same eigenvalues, since eigenvalues of a sum generally isn't a nice function of the eigenvalues of the summands. If this is not the case, what restrictions can we put on $A$, $B$, and $C$ to make this true?