Putnam 1990 B3

Let $S$ be a set of $2 \times 2$ integer matrices whose entries $a_{ij}$ (1) are all squares of integers and, (2) satisfy $a_{ij} \leq 200$. Show that if $S$ has more than 50387 ($= 15^4 - 15^2 - 15 + 2$) elements, then it has two elements that commute.