Jacobi's Determinant Identity


A=[B D; E C]
A^(-1)=[W X; Y Z],

where B and W are k×k matrices. Then


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

 [B D; E C][I X; 0 Z]=[B 0; E I],

where I is the identity matrix and 0 is the zero matrix.

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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.

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Jacobi's Determinant Identity

Cite this as:

Weisstein, Eric W. "Jacobi's Determinant Identity." From MathWorld--A Wolfram Web Resource.

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