Similarity of $2X2$ matrix when it has exactly one eigenvalue.

Question

I'm having a difficulty to determine if the following is true or false.

Let $A \in M_{nxn}(\mathbb{R}).$ If $n=2$ and $A$ has exactly one eigenvalue, then $A$ is similar to $\begin{pmatrix}\lambda &0\\ \:0&\lambda \:\end{pmatrix}$ or $\begin{pmatrix}\lambda &1\\ \:0&\lambda \:\end{pmatrix}$.

Is there a counter example?