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# Matrix Proofs : Linear Programming, Duality, Feasibility and Optimal Solutions

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Exercise 4.26 Let A be a given matrix. Show that exactly one of the following alternatives must hold.
(a) There exists some x does not equal 0 such that Ax = 0, x > 0.
(b) There exists some p such that p'A> 0'.
Exercise 4.27 Let A be a given matrix. Show that the following two statements are equivalent.
(a) Every vector such that Ax > 0 and x > 0 must satisfy x1 = 0.
(b) There exists some p such that p'A <0, p > 0, and p'A1 >0, where A1 is the first column of A.

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