Spectrum of a Hermitian matrix

Let $A=\begin{pmatrix}2 & 1-i\\ 1+i & 3\end{pmatrix}$. Check that $A=A^\dagger$, so its eigenvalues are real. Solve $\det(A-\lambda I)=0$ and enter the two eigenvalues $\lambda_1,\lambda_2$ as real numbers, in any order. Answers are accepted within $10^{-3}$.

Find the eigenvector $v=(v_1,v_2)^T$ of $A$ with $Av=4v$, normalised so that $\lVert v\rVert^2=\lvert v_1\rvert^2+\lvert v_2\rvert^2=1$ and with the global phase fixed so that $v_1$ is real and positive. Enter $v_1$ and $v_2$ as complex numbers. Answers are accepted within $2\times10^{-3}$.