Question: Adjoint Matrix formulated as permuted Hadamard products needs a parity matrix only for even dimensions. Why?

For a project I need to construct a large symbolic adjoint matrix, hoping it can be factored afterward into nice expressions.
In the worksheet, I present an adjoint matrix using permuted Hadamard products. What puzzles me is that only for even dimensions, I need to multiply it (elementwise) with a parity matrix. Okay, not a specific Maple question, but maybe someone can help me out.


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