Jacobi's Determinant Identity

Let

$\displaystyle {\hbox{\sf A}}$	$\textstyle =$	$\displaystyle \left[\begin{array}{cc}{\hbox{\sf B}}& {\hbox{\sf D}}\\ {\hbox{\sf E}} & {\hbox{\sf C}}\end{array}\right]$	(1)
$\displaystyle {\hbox{\sf A}}^{-1}$	$\textstyle =$	$\displaystyle \left[\begin{array}{cc}{\hbox{\sf W}} & {\hbox{\sf X}}\\ {\hbox{\sf Y}} & {\hbox{\sf Z}}\end{array}\right],$	(2)

where B and ${\hbox{\sf W}}$ are $k\times k$ Matrices. Then

$\begin{displaymath} ({\rm det\ }{\hbox{\sf Z}})({\rm det\ }{\hbox{\sf A}})={\rm det\ }{\hbox{\sf B}}. \end{displaymath}$

(3)

The proof follows from equating determinants on the two sides of the block matrices

$\begin{displaymath} \left[{\matrix{{\hbox{\sf B}}& {\hbox{\sf D}}\cr {\hbox{\sf ... ...{\hbox{\sf O}}\cr {\hbox{\sf E}} & {\hbox{\sf I}}\cr}}\right], \end{displaymath}$

(4)

where I is the Identity Matrix and O is the zero matrix.

References

Gantmacher, F. R. The Theory of Matrices, Vol. 1. New York: Chelsea, p. 21, 1960.

Horn, R. A. and Johnson, C. R. Matrix Analysis. Cambridge, England: Cambridge University Press, p. 21, 1985.