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 $\det M\ne0$; choosing the dimension alone does not ensure invertibility.

Solution

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 / v / Solution

 Solution (source code)