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2015 HMIC #3: Linear Algebra!?!?

Source:

May 11, 2015
HMIC

Problem Statement

Let MM be a 2014×20142014\times 2014 invertible matrix, and let F(M)\mathcal{F}(M) denote the set of matrices whose rows are a permutation of the rows of MM. Find the number of matrices F∈F(M)F\in\mathcal{F}(M) such that det⁡(M+F)≠0\det(M + F) \ne 0.
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