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What's the sufficient and necessary condition for $\det(I+\Omega A)=\det(I-\Omega B)$? Here, all the matrices are $3\times3$, $\Omega$ is skew-symmetric with its determinant equals to zero.

If the sufficient and necessary condition is difficult to find, what's the sufficient condition for this equation to be true? I know one of them is $A^T=A=B$ but it's too strict for my application.

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