Complex eigenvalues and eigenvectors

Find the (real or complex) eigenvalues and eigenvectors of the following matrices. Diagonalize each matrix if possible.

I know a certain real $2\times 2$-matrix $A$. My matrix has complex eigenvalue $\lambda=1+2i$ and corresponding eigenvector $\mathbf{v}=\left[\begin{array}{c}1\\ i\end{array}\right]$.

1. Find another eigenvalue and corresponding eigenvector of $A$.

2. Diagonalize $A$.

3. What is the secret matrix $A$?