$a_{ii} = \lambda_1$ for some $i$, then every other entry of row and column $i$ is zero.

Question

Let $A$ be an $n\times n$ Hermitian matrix with eigenvalues $\lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n$. Prove that if $a_{ii} = \lambda_1$ for some $i$, then every other entry of row and column $i$ is zero. Simillarly prove that if $a_{ii} = \lambda_n$ for some $i$, then every other entry of row and column $i$ is zero.

Need some hints to solve the problem.