Let $E\to S^1$ be a real rank-$k$ vector bundle described by a clutching matrix $A\in GL(k,\mathbb R)$ after cutting $S^1$ into an interval and gluing the two endpoint fibres by the relation $(1,v)\sim(0,Av)$ for every $v\in\mathbb R^k$. If $\det A>0$, then $E$ is orientable; if $\det A<0$, then $E$ is not orientable.