block determinants

If $A$ and $D$ are square matrices

• If $A^{-1}$ exists, then

• If $D^{-1}$ exists, then

The matrices $D-C{A}^{-1}B$ and $A-B{D}^{-1}C$ are called the Schur complements of $A$ and $D$, respectively.
Mention that

• If $A$, $D$ are square matrices, then

  , where $O$ is a zero matrix.

• Also we have that

• Another useful result for block determinants is the following.
  As is a symplectic matrix,we have that $detJ=1$. Using now the fact that $detMN=det(M)det(N)$ for any $M$, $N$ square matrices, we have that

This holds for any square matrices $A$, $B$ and for the last point $A$, $B$ have also the same order. They do not need to beinvertible.