Jacobi's formula

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

${\displaystyle {\frac {d}{dt}}\det A(t)=\operatorname {tr} \left(\operatorname {adj} (A(t))\,{\frac {dA(t)}{dt}}\right)=\left(\det A(t)\right)\cdot \operatorname {tr} \left(A(t)^{-1}\cdot \,{\frac {dA(t)}{dt}}\right)}$

where tr(X) is the trace of the matrix X and ${\displaystyle \operatorname {adj} (X)}$ is its adjugate matrix. (The latter equality only holds if A(t) is invertible.)

As a special case,

${\displaystyle {\partial \det(A) \over \partial A_{ij}}=\operatorname {adj} (A)_{ji}.}$

Equivalently, if dA stands for the differential of A, the general formula is

${\displaystyle d\det(A)=\operatorname {tr} (\operatorname {adj} (A)\,dA)=\det(A)\operatorname {tr} \left(A^{-1}dA\right)}$

The formula is named after the mathematician Carl Gustav Jacob Jacobi.