Can't get a lower triangular nxn determinant
The determinant $d$ is unchanged if you replace the first row by $$ \mathbf{r}_1 \leftarrow \mathbf{r}_1 - \frac{1}{a_1} \mathbf{r}_2 - \ldots - \frac{1}{a_N} \mathbf{r}_{N+1} $$ (if none of the $a$'s are null)
It follows that $$ d= \begin{vmatrix} -\sum a_n^{-1} & 0 & 0 & & 0\\ 1 & a_1 & & & \\ 1 & & a_2 & & \\ \vdots & & & \ddots & \\ 1 & & & & a_N \end{vmatrix}= -(\sum a_n^{-1})(\prod a_n) $$