Originally Posted by

**HallsofIvy** Why do you say that does not make sense? Why can the determinant not be 0? That matrix has eigenvalues 0 (because the determinant is 0), i and -i. That means each $\displaystyle x_i$ will be of the form A+ B cos(x)+ C sin(x).

But, personally, I wouldn't use a matrix solution. Differentiating the first equation again, we have $\displaystyle x_1''= x_3'= -x_1$ so that $\displaystyle x_1''+ x_1= 0$. That equation has general solution $\displaystyle x_1(t)= Acos(t)+ Bsin(t)$. Then $\displaystyle x_3= x_1'$ so that $\displaystyle x_3= -Asin(t)+ Bcos(t)$. Finally, $\displaystyle x_2'= 2x_1- x_3= (2A- B)cos(t)+ (2B+A)sin(t)$ so that $\displaystyle x_2= (2A- B)sin(t)- (2B+A)cos(t)+ C$ just as I said.