Show that the determinant is n factorial

Question

how can I prove the following problem?

$$\det\begin{bmatrix} 1 & 1 & 0 & 0 & \cdots & 0\\ -1 & 1 & 2^2 & 0 & \cdots & 0 \\ 0 & -1 & 1 & 3^2 & \ddots & \vdots\\ \vdots & \ddots & \ddots & \ddots & \ddots & 0\\ 0 & \cdots & 0 & -1 & 1 & (n-1)^2\\ 0 & \cdots & \cdots & 0 & -1 & 1\end{bmatrix} = n!$$ I'm familiar how to calculate a determinant, but how can I prove this?

Answer 1

Let $m=n-1$. The matrix has a very nice LU decomposition $$ \pmatrix{ 1\\ -1&2\\ &-1&3\\ &&\ddots&\ddots\\ &&&-1&m\\ &&&&-1&n} \pmatrix{ 1&1\\ &1&2\\ &&1&3\\ &&&\ddots&\ddots\\ &&&&1&m\\ &&&&&1}, $$ which makes its determinant obvious to calculate.