Let n ≥ 2. We look for three n x n matrices A,B,C over the non-negative integers with the following properties:

- The positions (1,1) and (2,2) are strictly positive in all three matrices A,B,C.
- We write X ≥ Y iff X
_{ij}≥ Y_{ij}for every i,j, and we write . for matrix multiplication. We require A.A ≥ B.C and B.B ≥ A.C and C.C ≥ A.B -
Among the three inequalities (A.A)
_{12}≥ (B.C)_{12}, (B.B)_{12}≥ (A.C)_{12}, (C.C)_{12}≥ (A.B)_{12}at least one is strict.

Several techniques have found solutions of the above problem, but not really systematically, and all requiring quite some computation time. Moreover, we should like to know the smallest n for which there is a solution.

Hans Zantema (H.Zantema@tue.nl)

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