Solution
= Solution
The determinant $\det(M-tI)$ is a polynomial in $t$ with leading term $(-t)^n$, so it is not the zero polynomial and has only finitely many roots. Hence one may choose $\alpha>0$ smaller than every positive root, if any. Then
$$
\boxed{M-tI\text{ is nonsingular for }0<t<\alpha}.
$$