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Frobenius inequality states that $\operatorname{rank} AB + \operatorname{rank} BC \leq \operatorname{rank} ABC + \operatorname{rank} B$ 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. :(2011-10-15

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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)