= Solution
For $\lambda\ne0$, factor the matrix inside the <determinant>:
$$
\begin{aligned}
\chi_M(\lambda)
&=\det M\,\det(\lambda M^{-1}-I)\\
&=\lambda^{2m}\det(M^{-1}-\lambda^{-1}I)\\
&=\lambda^{2m}\det(\lambda^{-1}I-M^{-1})\\
&=\lambda^{2m}\chi_M(\lambda^{-1}).
\end{aligned}
$$
The second line uses $\det M=1$; the third uses the even matrix size; the last uses part (i). Consequently
$$
\boxed{\chi_M(\lambda)=\lambda^{2m}\chi_M(\lambda^{-1}).}
$$
In other words, the <characteristic polynomial> equals its <reciprocal polynomial>.
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