Solution for system of diff equations

Find the solution of the system

x' = 3x + 4y; y' = -2x-3y

satisfying x(0) = 2 and y(0) = -1

I turned the pair of equations into a single first order vector differential equation of the form

v' = Av

where v is a column vector with entries x and y and A is a 2x2 matrix.

A= 3 4

-2 -3

=>det(A-Ir),where r is a constant and I is the identety matrix....=(3-r)(-3-r)+8=r^2-1=>

r1=-1 and r2=1 these are the eigenvectors.Was my approach ok?

Finding egenvectors:

(A-Ir)w_j=|3-1 4 | w1 = 0

|-2 -3+1 | w2 0

this result the system 2w_1+4w_2=0 and -2w_1-2w_2=0 where both w's =0

so the general solution is u=a_1 e^t *|0| +a_2e^-t|0|

|0| |0|

I know i did a mistake here ,was r1=i and r2=-i? ..adn yes I have to use eigenvectors and eigenvalues