1
$\begingroup$

Here is the differential equation $$N'_x(x)=G(x)N(x)$$ where $N, G$ are $2\times2$ matrices depending on $x$, and $G$ satisfies $\mathrm{trace} (G)=0$. My question is:

How can one then calculate the determinant of $N$, since for $G$ not a constant, one cannot use an exponential of a matrix and the formula $\det e^{A}=e^{\mathrm{trace} (A)}$?

$\endgroup$
  • $\begingroup$ You can find the answer here (the determinant is constant for traceless $G$). $\endgroup$ – Start wearing purple May 29 '13 at 15:26
  • $\begingroup$ By the way, for the $2\times 2$ case, you can just write this down explicitly. Differentiate the determinant and use the differential equation to simplify to $0$. $\endgroup$ – Ted Shifrin May 29 '13 at 18:18
0
$\begingroup$

See http://en.wikipedia.org/wiki/Jacobi's_formula - using this together with the cyclic nature of the trace and properties of the adjugate should get you where you need.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.