Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 2 / 12G / c / Solution

The function $X\mapsto \det (I-X)$ is a polynomial in the matrix entries, hence continuous. Consequently $U$ is the inverse image of the open set $\mathbb{R}\setminus \{0\}$ and is an open set.

If $\Vert X\Vert <1$ and $(I-X)v=0$, then $v=Xv$ implies $\Vert v\Vert \le \Vert X\Vert \Vert v\Vert$. This forces $v=0$. The square matrix $I-X$ is therefore injective, hence an invertible matrix, soThe argument works with any of the induced operator norms in part (b).