Characteristic polynomial/Product rule/Wrong proof/Exercise

What is wrong in the following argumentation:

"For two ${}n\times n$-matrices ${}M,N$, the characteristic polynomials fulfill the relation

${}\chi _{M\circ N}=\chi _{M}\chi _{N}\,.$

This is because, by definition, we have

${}\chi _{M\circ N}=\det {\left(XE_{n}-M\circ N\right)}=\det {\left(XE_{n}-M\right)}\det {\left(XE_{n}-N\right)}=\chi _{M}\cdot \chi _{N}\,,$

where the equation in the middle rests on the multiplication theorem for determinants“.

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