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 $\operatorname{ rank}(M) \geq \bigl(\frac{ 1}{ 2} - o(1)\bigr)n$.
Categories: math.CO