While answering the question Trace of a real, symmetric positive semi-definite matrix, the OP asked a follow-up question, that I was not able to answer at that moment. However, I came up with an intriguing (at least to me) question.
Let $\mathcal{S_n}(\mathbb R)$ be the class of all the real $n\times n$, symmetric, positive semi - definite matrices, with one entry strictly greater than $1$. Then, is it true that for all $A\in \mathcal{S_n}(\mathbb R)$ holds $$\rho(A) \ge 1?$$
Symmetric positive definite matrices $\subset$ Hermitian matrices $\subset$ normal matrices $\subset$ radial matrices (the set of all matrices $A\in M_n(\mathbb C)$ such that $\rho(A)=\|A\|_2$).
So, in your question, $\rho(A)=\|A\|_2$ and some $|a_{ij}|>1$. Hence $\rho(A)=\|A\|_2\ge\|Ae_j\|_2\ge|a_{ij}|>1$.