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 $\det e^{A}=e^{\mathrm{trace} (A)}$?

  • $\begingroup$ 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

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.


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