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Putnam 1992 B6

Source: Putnam 1992

July 18, 2022
Putnamlinear algebramatrix

Problem Statement

Let MM be a set of real n×nn \times n matrices such that
i) In∈MI_{n} \in M, where InI_n is the identity matrix.
ii) If A∈MA\in M and B∈MB\in M, then either AB∈MAB\in M or −AB∈M-AB\in M, but not both
iii) If A∈MA\in M and B∈MB \in M, then either AB=BAAB=BA or AB=−BAAB=-BA.
iv) If A∈MA\in M and A≠InA \ne I_n, there is at least one B∈MB\in M such that AB=−BAAB=-BA.
Prove that MM contains at most n2n^2 matrices.
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