Eigenvalues and eigenvectors of a 2×2 matrix

Eigenvalues. $A$ is upper triangular with diagonal entries $3$ and $2$, so the eigenvalues are $\lambda_1=3$ and $\lambda_2=2$.

Eigenvector for $\lambda=3$.
$$A-3I = \begin{pmatrix} 0 & 1 \\ 0 & -1 \end{pmatrix}.$$
The kernel is given by $y=0$, so an eigenvector is $v_1=(1,0)$.

Eigenvector for $\lambda=2$.
$$A-2I = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}.$$
The kernel is given by $x+y=0$, so an eigenvector is $v_2=(1,-1)$.

Conclusion. Eigenvalues $3$ and $2$, with eigenvectors $(1,0)$ and $(1,-1)$.