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April 28th, 2016, 05:21 PM  #1 
Newbie Joined: Apr 2016 From: Japan Posts: 2 Thanks: 0  A formula using det & tr
I've found a formula on one of my notebooks. $\displaystyle \frac{d}{dt} \det X(t) = \det X(t) \tr( X^{1}(t) \frac{d}{dt} X(t) ) $ Does anyone knoe how to prove it? 
June 3rd, 2016, 04:58 PM  #2 
Member Joined: May 2013 Posts: 31 Thanks: 3 
This formula can be obtained from the Jacobi formula. An attempt would to use: 1) Jacobi's formula for express the derivative of the determinant of A in terms of the adjugate of A and the derivative of A: $\displaystyle \frac{d}{dt} \det A(t) = \mathrm{tr} \left (\mathrm{adj}(A(t)) \, \frac{dA(t)}{dt}\right )$ 2) Assuming that A is invertible, is true: $\displaystyle A^{1} = \det A^{1} adj A.$ See: https://en.wikipedia.org/wiki/Jacobi's_formula 
June 3rd, 2016, 09:48 PM  #3 
Newbie Joined: Apr 2016 From: Japan Posts: 2 Thanks: 0 
Now I can understand well. Thank you very much,Prokhartchin! 

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