TOPICS
Search

Jacobi's Determinant Identity


Let

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

where B and W are k×k matrices. 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.


Explore with Wolfram|Alpha

WolframAlpha

More things to try:

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.

Referenced on Wolfram|Alpha

Jacobi's Determinant Identity

Cite this as:

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

Subject classifications