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egyptian fractions

Source: SEEMOUS 2010 P4

June 17, 2021
matrixlinear algebra

Problem Statement

Suppose that AA and BB are n×nn\times n matrices with integer entries, and det⁡B≠0\det B\ne0. Prove that there exists m∈Nm\in\mathbb N such that the product AB−1AB^{-1} can be represented as AB−1=∑k=1mNk−1,AB^{-1}=\sum_{k=1}^mN_k^{-1},where NkN_k are n×nn\times n matrices with integer entries for all k=1,…,mk=1,\ldots,m, and Ni≠NjN_i\ne N_j for i≠ji\ne j.
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