2

Show that if $A \in M_{nxn}(k)$ and $det A \neq 0$ then for each matrix $B \in M_{nxn}(k)$: $AB$ and $BA$ are similar. Give an example $A,B$ for which $AB$ and $BA$ are not similar.

If $AB$ and $BA$ are similar then I have matrix $C$ for which $AB=C^{-1}(BA)C$. However I don't know how to prove that this matrix exist if I know only that $det A \neq 0$. I thought also about a fact that $tr(AB)=tr(BA)$ but I think it laso is unhelpfull for me.

Can you get me some tips which facts are helpful for me to do this task?

MP3129
  • 3,195

1 Answers1

4

Hint: Note that $$ A(BA)A^{-1}=AB. $$

For the second part, we should choose both $A$ and $B$ as non-invertible. (Because if one of them is invertible, then above statement shows that $AB$ is similar to $BA$.) A hint is to find $A,B$ such that $AB=O$, $BA\ne O$.

Myunghyun Song
  • 21,723
  • 2
  • 24
  • 60
  • Ok, thanks. In that case, in the second part of the task you just need to take matrix $ A $ for which $ det A = 0 $ and any matrix $ B $? – MP3129 Mar 01 '19 at 09:00
  • 1
    @MP3129 I've added some part explaining that. I hope this will help to find $A,B$. – Myunghyun Song Mar 01 '19 at 09:23