Calculation of a Jacobian determinant

Question

I want to calculate the Jacobian determinant of the function$$ f(x,z_1, \ldots, z_{n-1}) = x (z_1, \ldots, z_{n-1},1 - z_1 - \ldots - z_{n-1}), $$ i.e. $$ \begin{vmatrix} z_1 & x & 0 & \ldots & \ldots & \ldots & 0 \\ z_2 & 0 & x & 0 & \ldots & \ldots & 0 \\ z_3 & 0 & 0 & x & 0 & \ldots & 0 \\ \vdots & \vdots & & \ddots & \ddots & \ddots & 0 \\ \vdots & \vdots & & & \ddots & \ddots& 0 \\ z_{n-1} & 0 & \ldots & \ldots & \ldots & 0 & x \\ 1- z_1 - \ldots - z_{n-1} & -x & \ldots & \ldots & \ldots & \ldots & -x \end{vmatrix}. $$ For $n=2$ I get $-x = (-x)^1$ and for $n=3$ I get $x^2 = (-x)^2$. So my guess for the $n$-dimensional case ($n$ variables) would be $(-x)^{n-1}$. How can I formally calculate the determinant? I tried to use Laplace's formula for the first row, but I couldn't calculate the determinant of the two minors.

Answer 1

Hint:

adding the first $n-1$ rows to the last one, the determinant becomes really simple.