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A Beautiful 2 by 2 Matrix Identity

While working on a variation of the RcppDynProg algorithm we derived the following beautiful identity of 2 by 2 real matrices:

The superscript “top” denoting the transpose operation, the ||.||^2_2 denoting sum of squares norm, and the single |.| denoting determinant.

This is derived from one of the check equations for the Moore–Penrose inverse and we have details of the derivation here, and details of the messy algebra here.