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.
Derivation
Via matrix computation Theorem. (Jacobi's formula) For any differentiable map A from the real numbers to n × n matrices,
d det ( A ) = tr ( adj ( A ) d A ) . {\displaystyle d\det(A)=\operatorname {tr} (\operatorname {adj} (A)\,dA).}
Proof. Laplace's formula for the determinant of a matrix A can be stated as
det ( A ) = ∑ j A i j adj T ( A ) i j . {\displaystyle \det(A)=\sum _{j}A_{ij}\operatorname {adj} ^{\rm {T}}(A)_{ij}.}
Notice that the summation is performed over some arbitrary row i of the matrix. The determinant of A can be considered to be a function of the elements of A:
det ( A ) = F ( A 11 , A 12 , … , A 21 , A 22 , … , A n n ) {\displaystyle \det(A)=F\,(A_{11},A_{12},\ldots ,A_{21},A_{22},\ldots ,A_{nn})}
so that, by the chain rule, its differential is
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