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In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A.
If A is a differentiable map from the real numbers to n × n matrices, then
where tr(X) is the trace of the matrix X. (The latter equality only holds if A(t) is invertible.)
As a special case,
Equivalently, if dA stands for the differential of A, the general formula is
The formula is named after the mathematician Carl Gustav Jacob Jacobi.