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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

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

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  • $\begingroup$ Good work. Did you google "derivative of determinant"? You will find your answer if you do... $\endgroup$
    – runway44
    May 20, 2020 at 8:06
  • $\begingroup$ @runway44 I found Jacobi's formula but I don't know how to get the $dA$ out of the $\rm Tr()$ $\endgroup$
    – Museful
    May 20, 2020 at 8:08
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    $\begingroup$ 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). $\endgroup$
    – runway44
    May 20, 2020 at 8:09
  • $\begingroup$ @runway44 I see... In numerator layout format would it be $adj(A)$ or $adj^T(A)$? $\endgroup$
    – Museful
    May 20, 2020 at 8:12
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    $\begingroup$ You can simplify $\frac{1}{\det(A)} \operatorname{adj}(A) = A^{-1}$. (Maybe needs transposing depending on layout convention) $\endgroup$
    – Hyperplane
    May 20, 2020 at 8:18

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