Let be a diagonalizable matrix having only one eigenvalue .
By the defintion of diagonalizability, there exists an nonsingular matrix such that (where B is a diagonal matrix).
The diagonal entries of are the eigenvalues of ; therefore, (which is a scalar matrix).
This means that A is similar to . But the only matrix similar to a scalar matrix is itself. So A is a scalar matrix.
I coincidentally just proved the last part in another thread about 20 minutes ago.