Solution
= Solution
For $d(X)=X^n+a_{n-1}X^{n-1}+\cdots+a_0$, the <determinant> of its <companion matrix> is $(-1)^na_0$. Therefore
$$
\boxed{\alpha\text{ invertible}\iff a_0\ne0.}
$$
Thus \b[invertibility is readable from the constant coefficient], with no factorization or elimination.