To me, $\mathrm{Mat}\_{m \times n}$ is isomorphic to $\mathbb{R}^{mn}$, hence is boundaryless. But this disqualified the use of Sard's theorem in this question: [An exercise on Regular Value Theorem](https://math.stackexchange.com/questions/417901/an-exercise-on-regular-value-theorem). But it seems using Sard's theorem is on the right track.

Thank you~