Let $ \mathbb {F} $ F be a field and suppose $ \mathbf {a} := (a_1, a_2, \dotsc ) $ a := ( a 1 , a 2 , ... ) is a sequence of non-zero elements in $ \mathbb {F} $ F . For a tournament T on $ n $ n , associate the $ n \times n $ n × n symmetric matrix $ M_{T}(\mathbf {a}) $ M T ( a ) (resp. skew-symmetric matrix $ M_{T, \mathrm {skew}}(\mathbf {a}) $ M T , skew ( a ) ) with zero diagonal as follows: for i<j, if the edge ij is directed as $ i \to j $ i → j in T, then set $ M_{T}(\mathbf {a}) = a_i $ M T ( a ) = a i (resp.  $ M_{T, \mathrm {skew}}(\mathbf {a}) = a_i $ M T , skew ( a ) = a i ), else set $ M_{T}(\mathbf {a}) = a_j $ M T ( a ) = a j (resp.  $ M_{T, \mathrm {skew}}(\mathbf {a}) = a_j $ M T , skew ( a ) = a j ). Let $ \mathcal {M}_{n}(\mathbf {a}) $ M n ( a ) (resp.  $ \mathcal {M}_{n, \mathrm {skew}}(\mathbf {a}) $ M n , skew ( a ) ) be the family consisting of all the $ n \times n $ n × n symmetric matrices $ M_{T}(\mathbf {a}) $ M T ( a ) (resp. skew-symmetric matrices $ M_{T, \mathrm {skew}}(\mathbf {a}) $ M T , skew ( a ) ) as T varies over all tournaments on $ n $ n . We show that any matrix in $ \mathcal {M}_n(\mathbf {a}) $ M n ( a ) or $ \mathcal {M}_{n, \mathrm {skew}}(\mathbf {a}) $ M n , skew ( a ) corresponding to a transitive tournament has a rank at least n−1, and this is best possible. This settles in a strong form a conjecture posed in Balachandran et al. An ensemble of high-rank matrices arising from tournaments; Linear Algebra Appl. 2023;658:310-318. As a corollary, any matrix in these families has rank at least $ \lfloor \log _2(n) \rfloor $ ⌊ log 2 ⁡ ( n ) ⌋ .