Let $A$ and B be $n\times n$ matrices over a field $F$. Then prove or disprove
$(i)\, AB$ and $BA$ have same characteristic values.
$(ii)\, AB$ and $BA $ have same characteristic polynomial.
$(iii)\, AB$ and $BA $ have same minimal polynomial.
Proof of $(i)$
Let $c$ be an eigen value of AB and v be corresponding eigen vector. Then
AB(v)=cv, B(ABv)=B(cv), BA(Bv)=c(Bv),
this implies $Bv$ is an eigenvector of $BA$ corr. to eigen value $c$.
But this statement holds if $Bv$ is nonzero.
What if $Bv=0$?
$(ii)$ If $A$ is invertible then ($A$ inverse)$(AB)A= BA$, i.e. $AB $ and $BA$ are similar and hence have same charateristic polynomial.
Similar proof when $B$ is invertible.
But what if both $A$ and $B$ are not invertible?
I have no idea about $(iii)$.
Please help me to complete these proofs.