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. 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.