I'm considering an $n\times n$ matrix $T_\epsilon$ such that there exists a unique inverse for $\epsilon \in (-\delta,\delta)$ for some $\delta > 0$. I'm trying to check whether I've correctly determined the following:
Derivative of the determinant (property found online): \begin{align*} \frac{d}{d\epsilon}\det T_\epsilon &= \det T_\epsilon \textrm{tr} \left( T_\epsilon^{-1}\frac{d\,T_\epsilon}{d\epsilon} \right) \end{align*}
Derivative of cofactor (using $\frac{d(T_\epsilon^{-1})}{d\epsilon} = -T_\epsilon^{-1}\frac{d(T_\epsilon)}{d\epsilon}T_\epsilon^{-1}$): \begin{align*} \textrm{cof} \, T_\epsilon &= T_\epsilon^{-T} \det T_\epsilon \\ \frac{d}{d\epsilon} \textrm{cof} \, T_\epsilon &= \frac{d\,T_\epsilon^{-T}}{d\epsilon}\det T_\epsilon + T_\epsilon^{-T} \frac{d\det T_\epsilon}{d\epsilon} \\ &= -T_\epsilon^{-T}\frac{d(T_\epsilon)^T}{d\epsilon}T_\epsilon^{-T}\det T_\epsilon + T_\epsilon^{-T}\det T_\epsilon \textrm{tr} \left( T_\epsilon^{-1}\frac{d\,T_\epsilon}{d\epsilon} \right) \\ &= (\det T_\epsilon )T_\epsilon^{-T}\left( -\frac{d(T_\epsilon)^T}{d\epsilon}T_\epsilon^{-T} + \textrm{tr} \left( T_\epsilon^{-1}\frac{d\,T_\epsilon}{d\epsilon} \right) \right) \end{align*}
Derivative of trace of matrix squared: the trace is a linear operator so we can change the order of operations: \begin{align*} \frac{d}{d\epsilon} \textrm{tr}(T_\epsilon^2) &= \textrm{tr}\frac{d\,T_\epsilon^2}{d\epsilon} \\ &= \textrm{tr}\left(\frac{d\,T_\epsilon^T T_\epsilon}{d\epsilon} \right) \\ &= \textrm{tr}\left(\frac{d\,T_\epsilon^T}{d\epsilon}T_\epsilon + T_\epsilon^T \frac{d\,T_\epsilon}{d\epsilon} \right) \\ &= \textrm{tr}\left(2\frac{d\,T_\epsilon^T}{d\epsilon}T_\epsilon \right) \\ &= 2\textrm{tr}\left(\frac{d\,T_\epsilon^T}{d\epsilon}T_\epsilon \right) \end{align*}
Do these seem the right way to go about for solving? Thank you for any input!