Calculate the following determinant How do I calculate the following determinant?
$$
 \begin{vmatrix}
  a_n & -1 & 0 & \dots & 0 & 0 \\
  a_{n-1} & x & -1 & \dots & 0 & 0 \\
  a_{n-2} & 0 & x & \dots & 0 & 0 \\
  \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
  a_1 & 0 & 0 & \dots & x & -1 \\
  a_0 & 0 & 0 & \dots & 0 & x
 \end{vmatrix} 
$$

Let $F(a_0, a_1, \dots, a_n)$ be the determinant in your question. Expanding by minors across the first row, we get that $F(a_0, a_1, \dots, a_n)$ is equal to
$$
 a_n
 \begin{vmatrix}
  x & -1 & \dots & 0 & 0 \\
  0 & x  & \dots & 0 & 0 \\
  \vdots & \vdots & \ddots & \vdots & \vdots \\
  0 & 0 & \dots & x & -1 \\
  0 & 0 & \dots & 0 & x
 \end{vmatrix}
 -(-1)
  \begin{vmatrix}
  a_{n-1} & -1 & \dots & 0 & 0 \\
  a_{n-2} & x  & \dots & 0 & 0 \\
  \vdots & \vdots & \ddots & \vdots & \vdots \\
  a_1 & 0 & \dots & x & -1 \\
  a_0 & 0 & \dots & 0 & x
 \end{vmatrix}
$$
or $a_n x^n + F(a_0, a_1, \dots, a_{n-1}).$
From there, you can get the answer by induction on $n$.
