Is this answer on eigenvector diagonalisation wrong?

Question

This is the question and answer from MIT OCW 18.06 on eigenvectors and diagonalisation:

Two things I don't understand:

1. Shouldn't the eigenvector for $\lambda = -0.3$ be $\begin{bmatrix} -1 \\ 1 \end{bmatrix}$ because the second column is the free variable?
2. Assuming the answer is right in 1, wouldn't $S^{-1} = \frac{1}{3} \begin{bmatrix} -1 & -1 \\ -4 & 9 \end{bmatrix}$ because $\begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$?

Answer 1

1. It doesn't matter. $(1,-1)$ is as good an eigenvector as $(-1,1)$ or $(3,-3)$ are.

2. $ad-bc=-13$, therefore a minus sign got absorbed in $+\frac1{13}$.