Computing determinant of the matrix $C$

Question

Let

$$C=\begin{bmatrix} 0 & 0 & \cdots &0 & -c_0 \\ 1 & 0 & \cdots & 0& -c_1 \\ 0& 1 & \cdots & 0& -c_2 \\ \vdots & \vdots & & & \\ 0 & 0 & \cdots & 1 & -c_{n-1} \end{bmatrix}$$

Then why

$$\det (zI-C) = c_0 + c_1 z + \ldots + c_{n-1}z^{n-1} + z^n$$

?

Answer 1

Hint: Use Laplace expansion in the last column.