Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2003 / Paper 1 / 5 / c / i / Solution

The product of a $3\times n$ matrix with another $3\times n$ matrix requires its inner dimensions $n$ and three to match. Thus $\boxed{n=3}$ is necessary and sufficient for $M^2$ to be defined.