We have the following linear homogenous DE system
$X' = AX, \tag 0$
I wanna prove with induction that
$dW/dx = Tr(A)*W$
So for n=2
based on the above, we get,
$A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}, \tag 1$
and
$X = \begin{bmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{bmatrix}, \tag 2$
we have
$X' = \begin{bmatrix} x_{11}' & x_{12}' \\ x_{21}' & x_{22}' \end{bmatrix} = AX = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}\begin{bmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{bmatrix} = \begin{bmatrix} a_{11}x_{11} + a_{12}x_{21} & a_{11}x_{12} + a_{12}x_{22} \\ a_{21}x_{11} + a_{22}x_{21} & a_{21}x_{12} + a_{22}x_{22} \end{bmatrix}; \tag 3$
we define the Wronskian
$W = \det \left ( \begin{bmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{bmatrix} \right ) = x_{11}x_{22} - x_{12}x_{21}; \tag 4$
we compute, using (3):
$W' = x_{11}'x_{22} + x_{11} x_{22}' - x_{12}'x_{21} - x_{12} x_{21}'$ $= (a_{11}x_{11} + a_{12}x_{21})x_{22} + x_{11}(a_{21}x_{12} + a_{22}x_{22}) - (a_{11}x_{12} + a_{12}x_{22})x_{21} - x_{12}(a_{21}x_{11} + a_{22}x_{21})$ $= a_{11}x_{11}x_{22} + a_{22}x_{11}x_{22} - a_{11}x_{12}x_{21} - a_{22}x_{12}x_{21}$ $= a_{11}(x_{11}x_{22} - x_{12}x_{21}) + a_{22}(x_{11}x_{22} - x_{12}x_{21}) = (a_{11} + a_{22})(x_{11}x_{22} - x_{12}x_{21}) = \text{Tr}(A) W. \tag 5$
But i cannot figure out the induction step. I would really apreciate some help. Thanks.
I'm not sure if I have to use induction method.
If you are allowed to use Jacobi's formula things will be even more straight forward. $$ \frac{d\det X(t)}{dt}=\det X(t) tr(X(t)^{-1}\frac{dX(t)}{dt})\\ =\det X(t) tr(X(t)^{-1}AX(t))\\ =\det X(t) tr(AX(t)X(t)^{-1})\\ =tr(A)\det X(t) $$
If Jacobi's formula is not allowed, here is a limit that could prove the target formula $$ \frac{d\det X(t)}{dt}=\lim_{\epsilon\to 0 } \frac{\det X(t+\epsilon)-\det X(t)}{\epsilon}\\ =\lim_{\epsilon\to 0 }\frac{\det (X(t)+AX(t)\epsilon + o(\epsilon^2))-\det X(t)}{\epsilon} $$ In the following I'll abbreviate $X(t)$ as $X$ $$ \frac{d\det X}{dt}=\lim_{\epsilon\to 0 }\frac{\det ((I+\epsilon A)X)-\det X}{\epsilon}\\ =\lim_{\epsilon\to 0 }\frac{\det (I+\epsilon A)-1}{\epsilon}\det X\\ =\lim_{\epsilon\to 0 }\frac{\prod_i (1+\epsilon A_{ii})+o(\epsilon^2)-1}{\epsilon}\det X\\ =\lim_{\epsilon\to 0 }\frac{1 + \epsilon\sum_i A_{ii}+o(\epsilon^2)-1}{\epsilon}\det X\\ =tr(A) \det X $$ In this process, I discard terms that are proportional to $\epsilon^2$ (denoted as $o(\epsilon^2)$) since they will vanish in the limit of the quotient.