How to find following $m$ order determinant?
$\begin{vmatrix} 1&1&1&1&1&\cdots&1\\ 1&-1&0&0&0&\cdots&0\\ 1&0&-1&0&0&\cdots&0\\ 1&0&0&-1&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\cdots&\vdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 1&0&0&0&0&\vdots&-1\\ \end{vmatrix}_m$
Recall that the determinant does not change if we add one row to another.
By adding all the rows to the first row, we obtain a diagonal matrix whose determinant is easy to evaluate
$\begin{align} \begin{vmatrix} 1&1&1&1&1&\cdots&1\\ 1&-1&0&0&0&\cdots&0\\ 1&0&-1&0&0&\cdots&0\\ 1&0&0&-1&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\cdots&\vdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 1&0&0&0&0&\vdots&-1\\ \end{vmatrix}_m &= \begin{vmatrix} m&0&0&0&0&\cdots&0\\ 1&-1&0&0&0&\cdots&0\\ 1&0&-1&0&0&\cdots&0\\ 1&0&0&-1&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\cdots&\vdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 1&0&0&0&0&\vdots&-1\end{vmatrix}_m\\\\&=(-1)^{m-1}m\end{align}\\ $
