Determinant of $N \times\ N$ matrix So the question asks: 

For $n \geq 2$, compute the determinant of the following matrix:
  $$
 B =
 \begin{bmatrix}
  -X     &  1     & 0      & \cdots &  0      & 0      \\
   0     & -X     & 1      & \ddots & \vdots  & \vdots \\
  \vdots & \ddots & \ddots & \ddots &  0      & \vdots \\
  \vdots &        & \ddots & \ddots &  1      & 0      \\
  0      & \cdots & \cdots & 0      & -X      & 1      \\
  a_0    & a_1    & \cdots & \cdots & a_{n-2} & (a_{n-1} - X)
 \end{bmatrix}
$$

Looking at the $2 \times 2$ and $3 \times 3$ forms of this matrix:
$\det \begin{bmatrix} -X & 0 \\ 0 & (a_1-X) \end{bmatrix} = -X(a_1-X) - 0 = X^2 - a_1X $
by expansion along the first row:
$\det \begin{bmatrix} -X & 1 & 0  \\ 0 & -X & 0 \\ 0 & 0 & (a_2-X) \end{bmatrix} = (-X) \times\det \begin{bmatrix} -X & 0 \\ 0 & a_2-X \end{bmatrix} - 1  \det\begin{bmatrix} 0 & 0 \\ 0 & a_2-X \end{bmatrix}$
$= (-X)[(-X)(a_2-X) -0] - 0 = X^3 - a_2X^2  $
So it looks like:
$\det  \begin{bmatrix}
  -X     &  1     & 0      & \cdots &  0      & 0      \\
   0     & -X     & 1      & \ddots & \vdots  & \vdots \\
  \vdots & \ddots & \ddots & \ddots &  0      & \vdots \\
  \vdots &        & \ddots & \ddots &  1      & 0      \\
  0      & \cdots & \cdots & 0      & -X      & 1      \\
  a_0    & a_1    & \cdots & \cdots & a_{n-2} & (a_{n-1} - X)
 \end{bmatrix}
= X^{n} - a_{n-1}X^{n-1} - a_{n-2}X^{n-2}  ... - a_1X$
Does this look right? Is "prove by induction" valid to use here? 

To add a final touch somewhat as a synthesis of the (thorough) answers of @user5713492 and @akech: the global result is that the companion matrix of polynomial p(X) is diagonalized with a Vandermonde matrix V(r_1,r_2,\cdots r_n) where the $r_k$ are the roots of p(X)" see https://en.wikipedia.org/wiki/Vandermonde_matrix
