Proof that a matrix is similar to one the following matrices.

Question

I am required to prove the following statement: Let $A \in \Bbb M_2(\Bbb R)$ . Prove that if $A$ has one eigenvalue λ, then A is similar to $ \begin{bmatrix} λ & 0 \\ 0 & λ \\ \end{bmatrix}$ or $ \begin{bmatrix} λ & 1 \\ 0 & λ \\ \end{bmatrix}$ I was able to prove that if the geometric multiplicity of λ is 2 then $A$ is diagonalizable so it is similar to $ \begin{bmatrix} λ & 0 \\ 0 & λ \\ \end{bmatrix}$. I was trying to find a way to prove that if the geometric multiplicity of λ is 1 then $A$ is similar to $\begin{bmatrix} λ & 1 \\ 0 & λ \\ \end{bmatrix}$ but couldn't figure out how. Assistance will be more than welcomed.