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Jacobi's Determinant Identity


$\displaystyle {\hbox{\sf A}}$ $\textstyle =$ $\displaystyle \left[\begin{array}{cc}{\hbox{\sf B}}& {\hbox{\sf D}}\\  {\hbox{\sf E}} & {\hbox{\sf C}}\end{array}\right]$ (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
({\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
\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.


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.

© 1996-9 Eric W. Weisstein