block determinants

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If $A^{-1}$ exists, then

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

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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.
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Also we have that

$\det\begin{pmatrix}A&O\\ O&B\end{pmatrix}=\det(A)\det(B).$
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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
Canonical name	BlockDeterminants
Date of creation	2013-03-22 15:25:57
Last modified on	2013-03-22 15:25:57
Owner	georgiosl (7242)
Last modified by	georgiosl (7242)
Numerical id	20
Author	georgiosl (7242)
Entry type	Theorem
Classification	msc 15A15
Related topic	SchurComplement
Related topic	DeterminantsOfSomeMatricesOfSpecialForm