Given $A$ and $B$ are matrices over finite field $\mathbb{Z}_p$ ($p$ is a prime number), does this statement holds ? $$\operatorname{rank} A + \operatorname{rank}B \leq \operatorname{rank} AB + n.$$
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2Possible duplicate of Sylvester rank inequality: $\operatorname{rank} A + \operatorname{rank}B \leq \operatorname{rank} AB + n$ ; the proofs given there work in any field. – Arnaud D. Nov 27 '19 at 16:06
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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.
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Dietrich Burde
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What about Frobenius Inequality ? Does it holds for a matrix over finite field ? – Aditya Aug 18 '17 at 14:49
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1This is why you should learn linear algebra from a proper text that doesn't dumb it down to real numbers. – darij grinberg Aug 18 '17 at 16:39