Jacobi's formula

Formula for the derivative of a matrix determinant / From Wikipedia, the free encyclopedia

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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.[1]

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.