Putnam 1992 B6

Let ${\mathcal{M}}$ be a set of real $n \times n$ matrices such that

(i) $I \in {\mathcal{M}}$, where $I$ is the $n \times n$ identity matrix;

(ii) if $A \in {\mathcal{M}}$ and $B \in {\mathcal{M}}$, then either $AB \in {\mathcal{M}}$ or $-AB \in {\mathcal{M}}$, but not both;

(iii) if $A \in {\mathcal{M}}$ and $B \in {\mathcal{M}}$, then either $AB = BA$ or $AB = -BA$;

(iv) if $A \in {\mathcal{M}}$ and $A \neq I$, there is at least one $B \in {\mathcal{M}}$ such that $AB = - BA$.

Prove that ${\mathcal{M}}$ contains at most $n^2$ matrices.