/sci/ - Science & Math

Putnam/Olympiad problem of the day from
http://www.math.harvard.edu/putnam/

Let M be a set of real n x n matrices such that
(i) <span class="math">I \in M[/spoiler], where I is the n x n identity matrix;
(ii) if <span class="math">A \in M[/spoiler] and <span class="math">B \in M[/spoiler], then either <span class="math">AB \in M[/spoiler] or <span class="math">-AB \in M[/spoiler], but not both;
(iii) if <span class="math">A \in M[/spoiler] and <span class="math">B \in M[/spoiler], then either AB = BA or AB = -BA;
(iv) if <span class="math">A \in M[/spoiler] and <span class="math">A \rlap{\kern{1.5pt}/}= I[/spoiler], there is at least one <span class="math">B \in M[/spoiler] such that AB = -BA.
Prove that M contains at most <span class="math">n^2[/spoiler] matrices.