Let $A$ be a non-scalar $2 \times 2$ complex matrix with a single eigenvalue $\lambda$. I want to show that $A$ is non-diagonalisable.
For $A$ to be diagonalisable, it must be that the geometric multiplicity of $\lambda$ is 2. This means that the eigenspace associated to $\lambda$ has dimension 2, and that there are two linearly independent eigenvectors associated to $\lambda$.
The problem is that I have no good idea about how to apply the fact that $A$ is non-scalar, since this is necessary to prove the statement.