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\end{equation}  This likely works because for any 2x2 matrix $|A|=-|AJ_2|=-|J_2A|=|J_2AJ_2|$.  This property $|A|=-|AJ_2|$ also appears to hold for a 3x3 matrix.  Extrapolating backwards for a two by two matrix we get the correct formula on the proviso we definfe $J_1\equiv-1$. This makes some sense, as for any $J_nJ_n=I$ and $J_nJ_nA=A$.