Show that a diagonalizable matrix having only one eigenvalue is a scalar matrix.
Let $\displaystyle A$ be a diagonalizable matrix having only one eigenvalue $\displaystyle \lambda $.
By the defintion of diagonalizability, there exists an nonsingular matrix $\displaystyle P$ such that $\displaystyle B = P^{-1}AP$ (where B is a diagonal matrix).
The diagonal entries of $\displaystyle B$ are the eigenvalues of $\displaystyle A$; therefore, $\displaystyle B = \lambda I $ (which is a scalar matrix).
This means that A is similar to $\displaystyle \lambda I $. 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.