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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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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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  • $\begingroup$ What about Frobenius Inequality ? Does it holds for a matrix over finite field ? $\endgroup$
    – Aditya
    Aug 18, 2017 at 14:49
  • $\begingroup$ This is also answered in the linked paper! $\endgroup$ Aug 18, 2017 at 14:52
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    $\begingroup$ This is why you should learn linear algebra from a proper text that doesn't dumb it down to real numbers. $\endgroup$ Aug 18, 2017 at 16:39

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