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

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.

The transpose has shape $n\times 3$. Multiplying two copies requires $3=n$, so $\boxed{n=3}$.

The shapes in $M{M}^T$ are $(3\times n)(n\times 3)$, whose inner dimensions always match. Hence every positive integer $n$ is allowed, and the resulting matrix is $3\times 3$.

Matrix addition requires equal shapes. The shapes $3\times n$ and $n\times 3$ agree exactly when $\boxed{n=3}$.

A two-sided matrix inverse requires a square matrix, so $\boxed{n=3}$ is necessary. It exists only if that square matrix is also an invertible matrix, equivalently $\det M\ne 0$; choosing the dimension alone does not ensure invertibility.