Jacobi's formula

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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 d d t det A ( t ) = tr ⁡ ( adj ⁡ ( A ( t ) ) d A ( t ) d t ) = ( det A ( t ) ) ⋅ tr ⁡ ( A ( t ) − 1 ⋅ d A ( t ) d t ) {\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 adj ⁡ ( X ) {\displaystyle \operatorname {adj} (X)} is its adjugate matrix. (The latter equality only holds if A(t) is invertible.) As a special case, ∂ det ( A ) ∂ A i j = adj ⁡ ( A ) j i = adj ⁡ ( A ) i j T ⟹ ∂ det ( A ) ∂ A = det ⁡ ( A ) A − T if A invertible. {\displaystyle {\partial \det(A) \over \partial A_{ij}}=\operatorname {adj} (A)_{ji}=\operatorname {adj} (A)_{ij}^{T}\Longrightarrow {\partial \det(A) \over \partial A}=\operatorname {det} (A)A^{-T}{\text{ if A invertible.}}} Equivalently, if dA stands for the differential of A, the general formula is d det ( A ) = tr ⁡ ( adj ⁡ ( A ) d A ) = det ( A ) tr ⁡ ( A − 1 d A ) {\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 Jacobi.