# Derivative of $\log^2(\det(A))$ w.r.t. matrix A

Given that $$A\in\mathbb{R}^{n\times n}$$, is there an explicit matrix expression for

$$\frac{d}{dA}\log^2(\det(A))$$

(in numerator layout format)?

### Attempt

$$=2\log(\det(A))\frac{d}{dA}\log(\det(A))$$ $$=2\log(\det(A))\frac{1}{\det(A)}\frac{d}{dA}\det(A)$$

• Good work. Did you google "derivative of determinant"? You will find your answer if you do... May 20, 2020 at 8:06
• @runway44 I found Jacobi's formula but I don't know how to get the $dA$ out of the $\rm Tr()$ May 20, 2020 at 8:08
• The second equation in Wikipedia's "Jacobi formula" gives a "special case" which is precisely $\frac{d}{dA}\det A$, no traces involved. It's the transpose of the adjugate matrix (so, the cofactor matrix). May 20, 2020 at 8:09
• @runway44 I see... In numerator layout format would it be $adj(A)$ or $adj^T(A)$? May 20, 2020 at 8:12
• You can simplify $\frac{1}{\det(A)} \operatorname{adj}(A) = A^{-1}$. (Maybe needs transposing depending on layout convention) May 20, 2020 at 8:18