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Given two vectors $$\vec{v} = \begin{pmatrix} v_1\\ \vdots\\ v_n \end{pmatrix} , \vec{w} = \begin{pmatrix} w_1\\ \vdots\\ w_n \end{pmatrix} \in \mathbb{R}^n$$

such that for all $1 \leq j, k \leq n$

  • $v_j \neq v_k $ for $j \neq k$
  • $w_j \neq w_k $ for $j \neq k$
  • $v_j \neq w_k $
  • $v_j >0 \quad $
  • $w_j >0 \quad $

Define a Matrix $A \in \mathbb{R^{n \times n}}$ with entries $a_{jk}$, for $1 \leq j, k \leq n$, by

$$a_{j k} :=\frac{1}{(v_j - w_k)^2}$$

Is there any simple way to get an expression for its inverse? I'm really a newbie in linear algebra, this object seems simple enough to be already known, but I can't solve this nor find it solved anywhere.
Thanks in advance! :)

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