# 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.