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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 define  $J_1\equiv-1$. This makes some sense, as for any $J_nJ_n=I$ and $J_nJ_nA=A$. We can further extrapolate to the inverse of a 1x1 matrix $A=A_{11}$, taking the $R_{11}$ element to be the zero matrix, the determinant of this matrix is $1$ and the reciprocal of the determinant of $A$ is then just the reciprocal of $A_{11}$, which again is the inverse of the 1x1 matrix.  Proof $J_1=-1$, for the property for $J_1=-1$:   For  any non-singular nxn matrix $A$ $J_n$, $n>1$, $|J_n|=-1$ as $J_n$ is defined to be an antidiagonal matrix.\\  $|AB|=|A||B|$. \\  Therefore $|AJ_n|=-|A|$. \\  If one extrapolates  tohold that $|A|=-|AJ_n|$, J_1, must equal zero. As  the determinant of a 1x1 matrix is just case $n=1$, For  the element. above to remain true, $J_1=-1$