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Frobenius inequality states that $\mathrm{rank} AB + \mathrm{rank} BC \leq \mathrm{rank} ABC + \mathrm{rank} B$ whenever this has a meaning. I remember being told that this was sometimes useful. Do you know of any example?

Thank you.

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Thanks but I can't read it. :( –  timofei Oct 15 '11 at 20:35
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The prove of Frobenius inequality plays very important role in linear algebra's problem. you also see PROBLEMS AND THEOREMS IN LINEAR ALGEBRA by V. Prasolov (www.amazon.com/Problems-Theorems-Translations-Mathematical-Monographs/dp/0821802364)

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