Show that $AB^{-1}$ and $B^{-1}A$ have the same eigenvalues 
Let $A$ and $ B$ be two nonsingular matrices. Show that $AB^{-1}$ and $B^{-1}A$ have the same eigenvalues

My attempt:
$$
\begin{align}
f(\lambda) &= | I\lambda -AB^{-1}|  \\
&= |(I\frac{1}{\lambda}-B^{-1}A)^{-1}| \\
&= \dfrac{1}{|I\frac{1}{\lambda}-B^{-1}A|} \\
\end{align}
$$
Not sure how to complete the final step. Any help would be appreciated
 A: For completeness, I'll expand on 5xum's comment.
Choose any eigenvalue $\lambda$ of $AB^{-1}$. Then we know that there exists some nonzero vector $x$ such that:
$$
(AB^{-1})x = \lambda x
$$
We want to show that $\lambda$ is also an eigenvalue of $B^{-1}A$. That is, we seek some nonzero vector $y$ such that:
$$
(B^{-1}A)y = \lambda y
$$
To this end, consider $y = B^{-1}x$ (which is a nonzero vector, since otherwise $x$ would be an eigenvector whose corresponding eigenvalue is $0$, contradicting the invertibility of $B^{-1}$). Observe that:
$$
(B^{-1}A)y = (B^{-1}A)(B^{-1}x) = B^{-1}((AB^{-1})x) = B^{-1}(\lambda x) = \lambda ( B^{-1}x) = \lambda y
$$
as desired. $~~\blacksquare$
A: Suppose $CD$ has a non-zero eigenvalue, then $CDv = \lambda v$ for some $v \neq 0$.
Then $D (CD)v = \lambda Dv = (DC) Dv$, and we know that $Dv \neq 0$ (otherwise $\lambda =0$), hence $\lambda$ is an eigenvalue of $DC$.
Since $A,B$ are non-singular, all eigenvalues of matrices $AB^{-1},B^{-1}A$ are non-zero, hence the have the same eigenvalues.
Alternatively,
\begin{eqnarray}
\det (sI-AB^{-1}) &=&\det ((sB-A) B^{-1}) \\
&=& \det (sB-A) \det B^{-1} \\
&=&  \det B^{-1} \det (sB-A) \\
&=& \det (B^{-1}(sB-A) ) \\
&=& \det (sI-B^{-1}A)
\end{eqnarray}
And finally, the simplest answer:
Since $AB^{-1}$ and $B^{-1} (A B^{-1}) B$ are similar matrices, they have exactly the same eigenvalues.
A: It's a bit tricky:
$$
\begin{align}
&f(\lambda)\\ 
&=\det(AB^{-1}-\lambda I_n)\\ 
&=\det((\color{#00A000}{BB^{-1}})AB^{-1}-\lambda (\color{#00A000}{BB^{-1}}))\\
&=\det(B(B^{-1}A)B^{-1}- B(\lambda I_n)B^{-1})\\
&=\det(\color{#00A000}B(B^{-1}A-\lambda I_n)\color{#00A000}{B^{-1}})\\
&=\det(BB^{-1})\det(B^{-1}A-\lambda I_n)\\
&=\det(B^{-1}A-\lambda I_n)
\end{align}
$$
In a more general fashion , you can relate the char polynomials of $AB$ and $BA$ even when $A$ and $B$ are not square matrices, see my answer.
