Most proofs of the characteristic polynomial of the companion matrix–an important specific case–proceed by induction, and start with a matrix. It strikes me that an inductive proof has more force (or at least makes more sense) if a larger matrix is used. In this case we will use a “large” (numerical analysts will laugh at this characterisation) matrix.
Let us begin by making a notation change. Consider the general polynomial
Our object is thus to prove that this (or a variation of this, as we will see) is the characteristic polynomial of
The characteristic polynomial of this is the determinant of the following:
To find the determinant, we expand along the first row. But then we discover that only two minors that matter: the one in the upper left corner and the one in the upper right. Breaking this up into minors and cofactors yields the following:
Repeating this process until the end, it is easy to see that
or more generally
where is the degree of the polynomial (and the size of the companion matrix.) If we drop the terms we used to make the polynomial monic, we have at last