Diagonalization & Matrix Powers · Challenge problem

A linear system $\mathbf x' = A\mathbf x$ has $A = \begin{pmatrix}3 & -6\\1 & -2\end{pmatrix}$, which has $0$ as one of its eigenvalues. This means there is a LEFT eigenvector $\mathbf w = (w_1, w_2)$ with $\mathbf wA = \mathbf 0$, making the quantity $q(t) = w_1x_1(t) + w_2x_2(t)$ constant in time (a conserved quantity). Normalizing $w_1 = 1$, find $w_2$.