So, I need to reduce the matrix $$A = \begin{bmatrix} 4 & -1 \\ 9 & -2 \end{bmatrix}$$ into its Jordan Canonical form. So, I need to find matrix $P$ such that $P^{-1}AP$ is a Jordan matrix.
So I found that $\lambda = 1$ so $v_1 = \begin{bmatrix} \frac{1}{3} \\ 1 \end{bmatrix}$. But now I need to find $v_2$ for the matrix $P$. I know that $P^{-1}AP$ should be $\begin{bmatrix} \ 1 & 1 \\ 0 & 1 \end{bmatrix}$. And I know that through solving $(A-\lambda I)x=v_1$ and letting $x=v_2$, I then got:
$x_1 = \frac{1}{9}+\frac{1}{3}t$ and $x_2=t$ but I'm not sure how to get $v_2$ out of this because everything that I try doesn't lead to the Jordan matrix.. Am I doing something wrong?