According to answer of Denis Serre to [this question](https://mathoverflow.net/questions/197773/a-geometric-property-of-singular-matrices), the manifold of singular matrices in $M\_{n}(\mathbb{R})$ is defined as follows:
$$M=\{A\in M\_{n}(\mathbb{R})\mid \text{rank}(A)=n-1\}$$


So we define a (line bundle) over this manifold:
$$\{(A,v)\in M\times\mathbb{R}^{n}\mid Av=0\}$$.



> 
> Is it a trivial line bundle?
> 
> 
> 



A line bundle over the manifold of singular matrices
No, this bundle is not trivial (starting from dimension $2$). Introduce a metric, consider projector to a hyperplane, and rotate this hyperplane through $\pi$ about an axis. You get an orientation reversing loop.


