For every matrix $A$, the matrix $$ex sex{a{cc} I&A\ 0&I ea} eex$$ is invertible and its inverse is $$ex sex{a{cc} I&-A\ 0&I ea}. eex$$ Use this to show that if $A,B$ are any two $n imes n$ matrices, then $$ex sex{a{cc} I&A\ 0&I ea}^{-1}sex{a{cc} AB&0\ B&0 ea} sex{a{cc} I&A\ 0&I ea}=sex{a{cc} 0&0\ B&BA ea}. eex$$ This implies that $AB$ and $BA$ have the same eigenvalues.(This last fact can be proved in another way as follows. If $B$ is invertible, then $AB=B^{-1}(BA)B$. So, $AB$ and $BA$ have the same eigenvalues. Since invertible matrices are dense in the space of matrices, and a general known fact in complex analysis is that the roots of a polynomial vary continuously with the coefficients, the above conclusion also holds in general.)
Solution. This follows from direct computations.