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If matrix $A$ in an LP (or $A^T$ in its dual) has full row (column- in dual) rank, is it possible that both primal and dual have multiple solutions?

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According to the table you have mentioned, there is no record for multiple implies multiple, every multiple solution will be expressed as degenerate in dual. Therefore, this is impossible that both have multiple solution.

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Thank you for your reply. The question now is if it is impossible to have multiple solutions for both primal and dual, why the first line of the table is not: --Multiple implies "Unique" and degenerate'-- (like the third line) – Quick Dec 15 '11 at 1:41

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