According to answer of Denis Serre to this question, 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?