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Jacobi's Determinant Identity

Jacobi's determinant identity relates a determinant to a complementary block of the matrix inverse. Let A be an n×n invertible matrix, partitioned as

A=[B D; E C]
(1)
A^(-1)=[W X; Y Z],
(2)

where B and W are k×k matrices, D and X are k×(n-k), E and Y are (n-k)×k, and C and Z are (n-k)×(n-k). Then

det(Z)det(A)=det(B).
(3)

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],
(4)

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

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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.

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Jacobi's Determinant Identity

Cite this as:

Weisstein, Eric W. "Jacobi's Determinant Identity." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/JacobisDeterminantIdentity.html

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