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April 28th, 2016, 05:21 PM   #1
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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?
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June 3rd, 2016, 04:58 PM   #2
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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
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June 3rd, 2016, 09:48 PM   #3
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Now I can understand well.
Thank you very much,Prokhartchin!
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