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**1. Homework Statement**

Show that for a second order cartesian tensor A, assumed invertible and dependent on t, the following holds:

## \frac{d}{dt} det(A) = det(a) Tr(A^{-1}\frac{dA}{dt}) ##

**2. Homework Equations**

## det(a) = \frac{1}{6} \epsilon_{ijk} \epsilon_{lmn} A_{il}A_{jm}A_{kn} ##

**3. The Attempt at a Solution**

My approach is the following:

we know that ## det(a) = \frac{1}{6} \epsilon_{ijk} \epsilon_{lmn} A_{il}A_{jm}A_{kn} ##

Applying the time derivative to this:

## \frac{d}{dt} det(A) = \frac{1}{6} \epsilon_{ijk} \epsilon_{lmn} (A_{il}'A_{jm}A_{kn}+ A_{il}A_{jm}'A_{kn}+ A_{il}A_{jm}A_{kn}') ##

## = det(A) (\frac{1}{A_{il}} A_{il}' +\frac{1}{A_{jm}} A_{jm}' + \frac{1}{A_{kn}} A_{kn}') ##

This is starting to look somewhat like the expression we look for, but from here on I am stuck. Any ideas on how to continue? Many Thanks!