Calculating eigenvalues and eigenvectors Question: Calculate the eigenvectors and eigenvalues of the following matrix $$\begin{pmatrix}3&-3\\0&-2\\ \end{pmatrix}$$
My attempt:
I have calculated the eigenvalues to be $\lambda = 3$ and $\lambda =-2$ and I have managed to get the eigen vector for $\lambda = 3$ to be \begin{pmatrix}1\\0\\ \end{pmatrix}
However I get \begin{pmatrix}1\\\frac{5}{3}\\ \end{pmatrix}
but the correct answer seems to be 
\begin{pmatrix}0\\1\\ \end{pmatrix}
Where am I going wrong?

The eigenvalues of a triangular matrix are simply the diagonal entries, so your eigenvalues $3$ and $-2$ are correct. As
$$\begin{pmatrix}
3 & -3 \\
0 & -2 \\
\end{pmatrix}
\begin{pmatrix}
1 \\ 0
\end{pmatrix}
= \begin{pmatrix}
3 \\ 0
\end{pmatrix}
= 3 
\begin{pmatrix}
1 \\ 0
\end{pmatrix},$$
the vector $\begin{pmatrix}1 \\ 0 \end{pmatrix}$ is an eigenvector corresponding to $\lambda = 3$.
As
$$\begin{pmatrix}
3 & -3 \\
0 & -2 \\
\end{pmatrix}
\begin{pmatrix}
1 \\ 5/3
\end{pmatrix}
= \begin{pmatrix}
-2 \\ -10/3
\end{pmatrix}
= -2 
\begin{pmatrix}
1 \\ 5/3
\end{pmatrix},$$
the vector $\begin{pmatrix}1 \\ 5/3\end{pmatrix}$ is an eigenvector corresponding to $\lambda = -2$.
So, your answers are correct. Note that $\begin{pmatrix}0 \\ 1\end{pmatrix}$ is not an eigenvector of this matrix, because
$$\begin{pmatrix}
3 & -3 \\
0 & -2 \\
\end{pmatrix}
\begin{pmatrix}0 \\ 1\end{pmatrix}
= \begin{pmatrix}-3 \\ -2\end{pmatrix}
$$
which is not a scalar multiple of $\begin{pmatrix}0 \\ 1\end{pmatrix}$.
