dx/dt = (-2 ,1;0,-2)x x(0)=(0;1)

So far i have eigen values of -2
Eigen vectors of V0=(0;1) and V1=(1;0)

and x(t)= C1e^-2t(0;1)+C2e^-2t(1;0)


Can anyone tell me if this is all correct for solving intial value problem?
also how am i able to check this if it is correct?