I just cannot get this thing to make the 2 by 2 matrices of letters I want. Help!Wait, fixed it myself. There is a thread in Meta about Latex/jsMath inconsistencies, one known problem is backslash being interpreted as an escape. So where I intended double backslash I just put three backslashes and that works for now. If it fails later I will switch to four or five backslashes.
Given a square matrix $M \in SO_n$ decomposed as illustrated (well, pretend ) with square blocks $A,D$ and rectangular blocks $B,C,$
$$M = \left( \begin{array}{cc}
A & B \\
C & D
\end{array} \right) ,$$
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, (I intend a multiplication of square matrices):
$$ \left( \begin{array}{cc} A & B \\ 0 & I \end{array} \right) \left( \begin{array}{cc} A^t & C^t \\ B^t & D^t \end{array} \right) = \left( \begin{array}{cc} I & 0 \\ B^t & D^t \end{array} \right). $$
