  Suppose $\mathbb{F}$ is a field and let $\mathbf{a} := (a_1, a_2, \dotsc)$ be
a sequence of non-zero elements in $\mathbb{F}$. For $\mathbf{a}_n := (a_1,
\dotsc, a_n)$, we consider the family $\mathcal{M}_n(\mathbf{a})$ of $n \times
n$ symmetric matrices $M$ over $\mathbb{F}$ with all diagonal entries zero and
the $(i, j)$th element of $M$ either $a_i$ or $a_j$ for $i < j$. In this short
paper, we show that all matrices in a certain subclass of
$\mathcal{M}_n(\mathbf{a})$ -- which can be naturally associated with
transitive tournaments -- have rank at least $\lfloor 2n/3 \rfloor - 1$. We
also show that if $\operatorname{char}(\mathbb{F}) \neq 2$ and $M$ is a matrix
chosen uniformly at random from $\mathcal{M}_n(\mathbf{a})$, then with high
probability $\operatorname{rank}(M) \geq \bigl(\frac{1}{2} - o(1)\bigr)n$.
