= Derivative of the determinant
{title2=$D(\det)_A$}
= Derivative of a determinant
{synonym}
For an invertible <matrix> $A$ and any <matrix> $H$,
$$
D(\det)_A(H)=\det(A)\operatorname{tr}(A^{-1}H).
$$
Indeed $\det(A+tH)=\det(A)\det(I+tA^{-1}H)$, and multilinearity shows that the linear coefficient of $\det(I+tB)$ is $\operatorname{tr}B$. At a singular <matrix> the <derivative> is instead written $\operatorname{tr}(\operatorname{adj}(A)H)$; this <polynomial> formula holds for all $A$.
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