How to solve a linear system like that

Question

I'm trying to reproduce the following result:

with
Solve[{ A1 == t/ϵ (B1 + B2 + B4 + B5), 
  B1 == t/ϵ (B2 + B3 + A1 + A5), 
  B2 == t/ϵ (B1 + B3 + A1 + A2), 
  B3 == t/ϵ (B1 + B2 + A2 + A5), 
  B4 == t/ϵ (B5 + B6 + A1 + A4), 
  B5 == t/ϵ (B4 + B6 + A1 + A3), 
  B6 == t/ϵ (B4 + B5 + A3 + A4)}, A1]

but without any result.

Michael E2 · Answer

It's sometimes hard to get a result in exactly the desired form, but this is close and equivalent:
Solve[{A1 == t/ϵ (B1 + B2 + B4 + B5), 
  B1 == t/ϵ (B2 + B3 + A1 + A5), 
  B2 == t/ϵ (B1 + B3 + A1 + A2), 
  B3 == t/ϵ (B1 + B2 + A2 + A5), 
  B4 == t/ϵ (B5 + B6 + A1 + A4), 
  B5 == t/ϵ (B4 + B6 + A1 + A3), 
  B6 == t/ϵ (B4 + B5 + A3 + A4)},
 {A1}, {B1, B2, B3, B4, B5, B6}]

(*  {{A1 -> -(((A2 + A3 + A4 + A5) t^2)/((3 t - ϵ) ϵ))}}  *)

cvgmt · Answer

Reduce can solve the equations,but the result is not so readable.
Here is another version of @Michael E2.
result = Eliminate[{A1 == t/ϵ (B1 + B2 + B4 + B5), 
   B1 == t/ϵ (B2 + B3 + A1 + A5), 
   B2 == t/ϵ (B1 + B3 + A1 + A2), 
   B3 == t/ϵ (B1 + B2 + A2 + A5), 
   B4 == t/ϵ (B5 + B6 + A1 + A4), 
   B5 == t/ϵ (B4 + B6 + A1 + A3), 
   B6 == t/ϵ (B4 + B5 + A3 + A4)}, {B1, B2, B3, B4, B5, B6}]
Reduce[result, A1]

How to solve a linear system like that

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