Trying to determine the determinant of an abstract matrix

Question

I'm trying to write about linear homogeneous recurrence relations and I've come up on the following matrix :$$A=\begin{pmatrix} c_1 & c_2 & \cdots & c_{k-1} & c_k\\ 1 & 0 & \cdots & 0 & 0\\ 0 & 1 & \ddots & 0 & 0\\ \vdots & \vdots & \ddots & \ddots & \vdots\\ 0 & 0 & \cdots & 1 & 0 \end{pmatrix},$$ where $c_1,\cdots c_k$ is a real number. I need to find the eigenvalues of this matrix. So far I've got that $$c_A(r)=\begin{vmatrix} c_1-t & c_2 & \cdots & c_{k-1} & c_k\\ 1 & -t & \cdots & 0 & 0\\ 0 & 1 & \ddots & 0 & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & 1 & -t \end{vmatrix},$$ I need to express it in its polynomial form but I can't. I've done it for $k=3$ and found $c_A=-r^3+c_1r^2+c_2r +c_3.$ I want to show that $c_A(r)= r^k - \sum_{i=1}^{k-1}c_{i}r^{k-i}$. Any help ?

Answer 1

Such a matrix (or rather, once we reversed the order of rows and columns, and possibly taken the transpose) is called a companion matrix.

The easiest way to compute the determinant is to expand along the last column. This gives $(-t)$ times the determinant of a smaller matrix of the same type, plus $(-1)^nc_k$. The result now follows by induction on $k$.