I just cannot get this thing to make the 2 by 2 matrices I want. Help!
Given a matrix $M \in SO_n$ decomposed as illustrated (well, pretend these are genuine 2 by 2 matrices) with square blocks $A,D$ and rectangular blocks $B,C,$
$$M \; = \; \pmatrix{ A & B \\ C & D },$$ then $\det A = \det D.$
What this says is that, in Riemannian geometry with an orientable manifold, the Hodge star operator is an isometry, a fact that has relevance for Poincare duality.
http://en.wikipedia.org/wiki/Hodge_duality
http://en.wikipedia.org/wiki/Poincar%C3%A9_duality
But the proof is a single line:
$$ \pmatrix{ A & B \\ 0 & I} \; \pmatrix{ A^t & C^t \\ B^t & D^t } \; \; = \; \;\pmatrix{ I & 0 \\ B^t & D^t }. $$