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Show that the following determinant is non-zero:

$$\overbrace{ \begin{vmatrix} 1 & 1 & \dots & 1\\ 2^{m+1}-1^{m+1} & 2^m-1^m & \dots & 2^1-1\\ 3^{m+1}-2^{m+1} & 3^m-2^m & \dots & 3^1-2^1\\ \vdots & \vdots & \ddots &\vdots \\ (m+1)^{m+1}-m^{m+1} & (m+1)^m-m^m & \dots & 1 \\ \end{vmatrix}}^{m+1 \text{ times}} \neq 0$$

I tried to proceed by induction but I couldn't find a way to reduce it to a smaller determinant of the same type. I don't know any other ways to proceed(maybe show it is invertible by construction of its inverse?) Any help would be appreciated. Thank you.

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    The use of elementary row operations will help make this into a [Vandermonde matrix](https://en.wikipedia.org/wiki/Vandermonde_matrix), and from there the nonzero determinant is shown by a cute proof that is worth remembering. – 2017-01-05

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