# block determinants

If $A$ and $D$ are square matrices

• If $A^{-1}$ exists, then

 $\det\begin{pmatrix}A&B\\ C&D\end{pmatrix}=\det(A)\det(D-CA^{-1}B)\\$
• If $D^{-1}$ exists, then

 $\det\begin{pmatrix}A&B\\ C&D\end{pmatrix}=\det(D)\det(A-BD^{-1}C)$

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

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

 $\det\begin{pmatrix}A&B\\ O&D\end{pmatrix}=\det(A)\det(D)$

, where $O$ is a zero matrix.

• Also we have that

 $\det\begin{pmatrix}A&O\\ O&B\end{pmatrix}=\det(A)\det(B).$
• Another useful result for block determinants is the following.
As $J=\begin{pmatrix}O&I\\ -I&O\end{pmatrix}$ is a symplectic matrix, we have that $\det J=1$. Using now the fact that $\det MN=\det(M)\det(N)$ for any $M$, $N$ square matrices, we have that

 $\det\begin{pmatrix}O&A\\ B&O\end{pmatrix}=\det\begin{pmatrix}O&A\\ B&O\end{pmatrix}\det J=-\det(A)\det(B)$

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 be invertible.

Title block determinants BlockDeterminants 2013-03-22 15:25:57 2013-03-22 15:25:57 georgiosl (7242) georgiosl (7242) 20 georgiosl (7242) Theorem msc 15A15 SchurComplement DeterminantsOfSomeMatricesOfSpecialForm