linear algebra - Find the determinant of $N times N$ matrix

I have the following $N \times N$ matrix.
$$\begin{vmatrix} 0 & 1 & 1 & \ldots & 1 \\ 1 & a_1 & 0 & \ldots & 0 \\ 1 & 0 & a_2 & \ldots & 0 \\ \vdots & \vdots& &\ddots& \vdots\\ 1 & 0 & 0 & \ldots & a_n \\ \end{vmatrix}$$

There seems to be a pattern going on for the determinant of the $5 \times 5$ version of this matrix, but I'm not sure how I would find the determinant for the $N \times N$ one.

Answer

Transform the matrix by the (determinant invariant) operation of adding $-a_i$ times the $(i+1)$th row on the first row. This gets us
$$\begin{vmatrix} -\sum_i \frac{1}{a_i} & 0 & 0 & \ldots & 0 \\ 1 & a_1 & 0 & \ldots & 0 \\ 1 & 0 & a_2 & \ldots & 0 \\ \vdots & \vdots& &\ddots& \vdots\\ 1 & 0 & 0 & \ldots & a_n \\ \end{vmatrix}$$
Then you have a lower triangle matrix whose determinant is just the product of the diagonal elements.