Let
 |  | ![[B D; E C]](/images/equations/JacobisDeterminantIdentity/Inline3.svg) |
(1)
|
 |  | ![[W X; Y Z],](/images/equations/JacobisDeterminantIdentity/Inline6.svg) |
(2)
|
where 
and 
are 
matrices. Then
 |
(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],](/images/equations/JacobisDeterminantIdentity/NumberedEquation2.svg) |
(4)
|
where 
is the identity matrix and 
is the zero matrix.
Jacobi's Determinant Identity
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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.Referenced on Wolfram|Alpha
Jacobi's Determinant IdentityCite this as:
Weisstein, Eric W. "Jacobi's Determinant Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JacobisDeterminantIdentity.html