Let $A$ and $B$ be $n \times n$ matrices over $\mathbb{C}$.

(a) Assuming that $A$ is invertible, show that $A B$ and $B A$ have the same characteristic polynomial.

(b) By considering the matrices $A-s I$, show that $A B$ and $B A$ have the same characteristic polynomial even when $A$ is singular.

(c) Give an example to show that the minimal polynomials $m_{A B}(t)$ and $m_{B A}(t)$ of $A B$ and $B A$ may be different.

(d) Show that $m_{A B}(t)$ and $m_{B A}(t)$ differ at most by a factor of $t$. Stating carefully any results which you use, deduce that if $A B$ is diagonalisable then so is $(B A)^{2}$.