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Let $A=[a_{ij}(x)]$ be a non singular matrix valued function with inverse $A^{-1}=B=[b_{ij}(x)]$

I am trying to use the chain rule to justify $\dfrac{\partial}{\partial x^i} (\log|\det A|)=\dfrac{(\operatorname{cof}A)_{rs}}{det A} \dfrac{\partial a_{rs}}{\partial x^i}=b_{rs} \dfrac{\partial a_{rs}}{\partial x^i}$

The solutions just say the proof follows by noting the expansion of determinant by rows $$\det A=\sum^n _{r=1} A_{ir} (\operatorname{cof} A)_{ir}$$ for any fixed $1 \leq i \leq n$ and then using the chain rule.

The proof I have from the book "Tensor calculus" by Schaums outline on page 106 is as given.

By the chain rule $$\frac{\partial}{\partial x^i} (\log |\det A|)=\frac{1}{\det A} \frac{\partial}{\partial x^i} (\det A)=\frac{1}{\det A} \frac{\partial}{\partial a_{rs}} (\det A) \frac{\partial a_{rs}}{\partial x^i}=\frac{A_{rs}}{\det A} \frac{\partial a_{rs}}{\partial x^i}=b_{sr} \frac{\partial a_{rs}}{ax^i}$$

Where does $\det A=\sum^n _{r=1} A_{ir} (\operatorname{cof} A)_{ir}$ come into it?

Al jabra
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  • When you're looking at $\frac{\partial}{\partial a_{rs}}(det A)$, notice that if you write out the determinant in terms of a cofactor expansion that include $a_{rs}$ in the row/column you're expanding in, then the cofactors are independent of $a_{rs}$, and the only term that survives is the cofactor of $a_{rs}$. Also when chain ruling from $x_i$ to $a_{rs}$ you should have a sum. – Alex R. Apr 29 '16 at 22:28
  • Applying the chain rule to $\det A=\sum^n {r=1} A{ir} (\operatorname{cof} A){ir}$ gives $\frac{d}{dt} \log \det A=\frac{\frac{d}{dt} det A}{det A}=\sum{i,j} \bigg(\frac{d}{dt} a_{ij} \bigg)$ – Al jabra Apr 30 '16 at 09:38
  • related? https://math.stackexchange.com/questions/1493137/prove-frac-partial-rmlnx-partial-x-2x-1-rmdiagx-1 – BCLC Apr 16 '21 at 10:23

1 Answers1

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The same question essentially was asked in Matrix identity involving trace and a final formula is $$d_t(\log\det A_t)=\operatorname{trace}\left(A_t^{-1}\;A_t^\prime\right)$$ where $t$ is your $x^i$. Of course, if you are actually asking about the derivative of the determinant map then you need to see the links therein, e.g. to Jacobi's formula.

Community
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rych
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  • related? https://math.stackexchange.com/questions/1493137/prove-frac-partial-rmlnx-partial-x-2x-1-rmdiagx-1 – BCLC Apr 16 '21 at 10:23