<s>[INST] Determinant $n\times n$ problem $$
D_n = \left| 
 \begin{matrix}
  n & -1 & -3 &  0 &  0 & \cdots & 0 & 0 &  0 \\
  n &  1 &  2 & -3 &  0 & \cdots & 0 & 0 &  0 \\
  n &  0 &  1 &  2 & -3 & \cdots & 0 & 0 &  0 \\
  n &  0 &  0 &  1 &  2 & \cdots & 0 & 0 &  0 \\
  n &  0 &  0 &  0 &  1 & \cdots & 0 & 0 &  0 \\
  \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots \\
  n &  0 &  0 &  0 &  0 & \cdots & 1 & 2 & -3 \\
  n &  0 &  0 &  0 &  0 & \cdots & 0 & 1 &  2 \\
  n &  0 &  0 &  0 &  0 & \cdots & 0 & 0 &  1 \\
 \end{matrix}
\right|
$$
Can anyone help me with this determinant? It is recursive formula but I can't get it.
 [/INST] Add the second up to the last rows to the first. Then the first row becomes $(n^2,0,\ldots,0)$. The new matrix is lower block-triangular and is in the form of $\pmatrix{n^2&0\\ n\mathbf1&A}$, where $A$ is by itself an upper triangular matrix with determinant 1. Hence $\det D_n=n^2$.
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