Iteratively define a matrix

It is easy with nicematrix.

\documentclass{article}
\usepackage{nicematrix}

\begin{document}
\NiceMatrixOptions{cell-space-top-limit = 2pt,cell-space-bottom-limit = 2pt}
\[
  F=\bAutoNiceMatrix{5-5}{\frac{G}{r^{2}_{\arabic{iRow}, \arabic{jCol}}}}
\]
\end{document}

A fairly general matrix generation macro:

\documentclass{article}
\usepackage{amsmath}
\usepackage{xparse,xfp}

\ExplSyntaxOn

\NewDocumentCommand{\generatematrix}{O{b}mmmO{0pt}}
 {% #1 = fence (default b)
  % #2 = n. of rows
  % #3 = n. of cols
  % #4 = entry template
  % #5 = extra spacing
  \sandalwood_matrix_generate:nnnnn { #1 } { #2 } { #3 } { #4 } { #5 }
 }

\tl_new:N \l__sandalwood_matrix_body_tl

\cs_new_protected:Nn \sandalwood_matrix_generate:nnnnn
 {
  \group_begin:
  \cs_set_protected:Nn \__sandalwood_matrix_entry:nn { #4 }
  \tl_clear:N \l__sandalwood_matrix_body_tl
  % generate the first m-1 rows, with the possible extra spacing in between
  \int_step_inline:nn { #2 - 1 }
   {
    % the first object in a row gobbles a spurious &
    \tl_put_right:Nn \l__sandalwood_matrix_body_tl { \use_none:n }
    % fill the row
    \int_step_inline:nn { #3 }
     {
      \tl_put_right:Nn \l__sandalwood_matrix_body_tl
       {
        & \__sandalwood_matrix_entry:nn { ##1 } { ####1 }
       }
     }
    % add the extra spacing at the end of a row
    \tl_put_right:Nn \l__sandalwood_matrix_body_tl { \\ \noalign{\vspace{#5}} }
   }
  % fill the last row
  \tl_put_right:Nn \l__sandalwood_matrix_body_tl { \use_none:n }
  \int_step_inline:nn { #3 }
   {
    \tl_put_right:Nn \l__sandalwood_matrix_body_tl
     {
      & \__sandalwood_matrix_entry:nn { #2 } { ##1 }
     }
   }
  \begin{#1matrix}
  \tl_use:N \l__sandalwood_matrix_body_tl
  \end{#1matrix}
  \group_end:
 }

\ExplSyntaxOff

\newcommand{\hilbertmatrix}[1]{%
  \generatematrix{#1}{#1}{\frac{1}{\inteval{##1+##2-1}}}[1ex]%
}

\begin{document}

\[
\generatematrix{5}{5}{\dfrac{G}{r_{#1#2}^2}}[1ex]=
\generatematrix[v]{6}{4}{a_{#1,#2}}+\hilbertmatrix{4}
\]

\end{document}

The leading optional argument (default b) is meant to state the shape of the fences, in the standard amsmath way. The trailing mandatory argument should be a length (extra space between rows).

The third mandatory argument to \generatematrix is a template, where #1 denotes the row index and #2 the column index. I added how to define a macro based on \generatematrix to show that if we use it in a definition, we need to transform them into ##1 and ##2.

^{I know that the operations don't make sense.}

This is the output of

\[
\generatematrix{2}{3}{\generatematrix[p]{2}{2}{A_{#1,##1}^{#2,##2}}[2pt]}[1ex]
\]