This is a relatively simple question and I've Google'd this topic but I can't seem to grasp the method. One site I visited helped be a bit, but it simply used substitution to solve the problem rather than elimination, so I feel as though it's very situational.
The question is the following:
The eigenvalues and eigenvectors of the matrix $\begin{bmatrix} 2 & -6 \\ 3 & -4 \end{bmatrix}$
They want me to diagonalize the matrix and find $S$ and $\Lambda$.
I have found the eigenvalues to be
$\Lambda = \begin{bmatrix} -1 + 3i & 0 \\ 0 & -1-3i \end{bmatrix} $
However, the method to find $S$, or the two eigenvectors, has me stuck. I've tried standard elimination. Using $-1 + 3i$, I got the following matrix after elimination
$\begin{bmatrix} 3+3i & -6 \\ 0 & 9-9i \end{bmatrix}$
I am not sure if I did the elimination incorrectly, but my process was to multiply the second row by $\frac{-3-3i}{3}$. Unfortunately, this isn't a singular matrix. Which step have I done wrong?