Given $A$ and $B$ are matrices over finite field $\mathbb{Z}_p$ ($p$ is a prime number), does this statement holds ? $$\operatorname{rank} A + \operatorname{rank}B \leq \operatorname{rank} AB + n.$$
2026-04-03 02:37:51.1775183871
Does Sylvester Rank Inequality holds for a matrix over finite field?
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Sylvester's rank inequalities hold over any field $K$. However, they may not hold over rings in general - see for example the article Rank inequalities over semirings and its references.