0 -2 has characteristic equation The matrix A det [0 -> a11 a12 = = a21 a22 1 -2 -2 = 1 -2-X (0-1)(-2-A) — (1)(−2) = - 12+2)+(2) = 0. The eigenvalue with positive imaginary part is 1 = -1+1 i For eigenvalue 1, the eigenvector has components a and b that satisfy (0-1)a+(-2)6 = 0, (1)a+(-21)b = 0 which is the system 1+-1 ia +-2b=0, 1a+1+i−1 ]b = 0. Choosing the first component of the eigenvector as a = 1, for 1 the eigenvector is a v = = b where b = +i The system of differential equations x' = 1 -2 - -2 has a general solution x(t) = = c1x1(t) + c2x2(t) where [Re (eta)] x1(t) = = [Re (etb)] [Im (e¹¹t a)] x2(t) = = Im (etb)] et cos(t) et sin(t) =