Give a Matrix $P \in M_{2,2}(\mathbb{C})$ with the following properties: i) the Matrix P has only real elements
ii) 2+i is an eigenvalue of Matrix P
I got that the zeros has to be $(x-(2+i))(x-(2-i))$ which is equal to $(x-2)^2 -i^2$ so the characteristical polynom is equal to $x^2-4x+4+1 =x^2-4x+5$ how can I find the matrix of this characteristical polynom?

So the zeros are $(x-(2+i))(x-(2-i))$ =
$$=((x-2)+i)((x-2)-i)$$
$$=(x-2)^2-i^2 = x^2-4x+4+1=x^2-4x+5$$
So we get that characteristical polynom has the factors $(2-x)(2-x)=x^2-2x-2x+4 +1$ which gives us the determinant:
\begin{vmatrix}
2-x & 1 \\ 
-1 & 2-x
\end{vmatrix}
the corresponding matrix is :\begin{pmatrix}
2 & 1\\
-1 & 2
\end{pmatrix}
or the determinant
\begin{vmatrix}
2-x & -1 \\ 
1 & 2-x
\end{vmatrix}
the corresponding matrix is :\begin{pmatrix}
2 & -1\\
1 & 2
\end{pmatrix}
