Let $A$ be a symmetric matrix of order $n$. If $\lambda_1,\ldots, \lambda_n$ are its eigenvalues and the main diagonal of $A$ is $\lambda_1, \ldots, \lambda_n$ then is $A$ diagonal?
If $n=2$, you can use the determinant to ensure that the nondiagonal entries are zero. If $n=3$ you can use the square of the trace. Is it possible to give a clean argument for a general $n$?