All the roots of characteristics polynomial of $AB$, where $A$, $B$ are skew symmetric matrices of order $2n$, are of multiplicity greater then $1$.

I know that eigenvalues of skew symmetric matrices are either $0$ or purely imaginary. Complex roots comes in pair, i.e., if $a$ is root of a real polynomial then $\bar{a}$ is a root as well.Also characteristic polynomial of $AB$ and $BA$ are same. These information are not enough to prove the above statement. Help me.

Thank you.


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